Exponent exercises
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This document provides examples and explanations of operations involving exponents, including: 1) Simplifying expressions with integer exponents and evaluating expressions with fractional exponents. 2) Moving terms with negative exponents to the other side of a fraction by taking their reciprocal. 3) Rewriting expressions with negative exponents as expressions with only positive exponents.
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4.2 Notes
1) The document discusses exponent rules including the zero exponent rule, expanded power rule, and negative exponent rules.
2) It provides examples of applying these rules such as 53=1, (3c2/2d3)4={81c8/16d12}, and (3/r3)-2=r6/9.
3) It lists group work problems assigned from sections 4.1 and 4.2 that are due on Wednesday.
Mathematics 9 Lesson 7: Laws of Exponents
This powerpoint presentation discusses or talks about the topic or lesson: Laws of Exponents. It also discusses and explains the rules, concepts, steps and examples of Laws of Exponents.
Laws of Exponents
This document discusses the laws of exponents. It defines exponents and provides examples of several exponent laws including:
1) The exponent of a number indicates how many times to multiply the base by itself.
2) Any base raised to the 0th power equals 1.
3) A negative exponent indicates taking the reciprocal of the base raised to the positive value of the exponent.
4) Laws for multiplying, dividing, distributing exponents across parentheses, and operating on quotients of expressions with the same exponent.
The distributive property
The document discusses the distributive property in mathematics. It defines the distributive property as the property that states if you divide both sides of an equation by the same nonzero number, the new equation will have the same solution. It provides examples of distributing multiplication over addition, such as 3(2 + 5) = 3*2 + 3*5. It also gives an example of distributing subtraction over multiplication, such as rewriting (16 - 7)^3 using the distributive property.
The distributive property (1)
The document discusses the distributive property in mathematics. It defines the distributive property as the property that states if you divide both sides of an equation by the same nonzero number, the new equation will have the same solution. It provides examples of distributing multiplication over addition, such as 3(2 + 5) = 3*2 + 3*5. It also gives an example of distributing subtraction over multiplication, such as rewriting (16 - 7)^3 using the distributive property.
Integral Exponents
This document discusses integral exponents and how to evaluate expressions with zero and negative exponents. It provides examples of simplifying expressions with zero and negative exponents by using the definition that a negative exponent means to take the reciprocal of the base and raise it to the positive value of the exponent. It also explains that any nonzero number raised to the zero power is equal to 1, and expressions should be rewritten with only positive exponents.
Exponents.Test.Slide.Show
Exponents describe the number of times a base is used as a factor. The base is the number being multiplied and the exponent shows how many times to multiply the base. For example, 2 to the 12th power means multiplying 2 by itself 12 times, or 2 x 2 x 2 x... = 4,096. Exponents allow large products to be written concisely using the format of base^exponent.
Exponent properties
This document summarizes exponent properties for adding, subtracting, multiplying, and dividing terms with exponents. It provides examples for each property:
1) When adding or subtracting terms with the same base and exponent, only the coefficient changes.
2) When multiplying terms, you add the exponents and multiply the coefficients.
3) When dividing terms, you subtract the bottom exponent from the top and divide the coefficients.
4) When simplifying powers of powers, you distribute the exponent and multiply the exponents.
Recommended
4.2 Notes
1) The document discusses exponent rules including the zero exponent rule, expanded power rule, and negative exponent rules.
2) It provides examples of applying these rules such as 53=1, (3c2/2d3)4={81c8/16d12}, and (3/r3)-2=r6/9.
3) It lists group work problems assigned from sections 4.1 and 4.2 that are due on Wednesday.
Mathematics 9 Lesson 7: Laws of Exponents
This powerpoint presentation discusses or talks about the topic or lesson: Laws of Exponents. It also discusses and explains the rules, concepts, steps and examples of Laws of Exponents.
Laws of Exponents
This document discusses the laws of exponents. It defines exponents and provides examples of several exponent laws including:
1) The exponent of a number indicates how many times to multiply the base by itself.
2) Any base raised to the 0th power equals 1.
3) A negative exponent indicates taking the reciprocal of the base raised to the positive value of the exponent.
4) Laws for multiplying, dividing, distributing exponents across parentheses, and operating on quotients of expressions with the same exponent.
The distributive property
The document discusses the distributive property in mathematics. It defines the distributive property as the property that states if you divide both sides of an equation by the same nonzero number, the new equation will have the same solution. It provides examples of distributing multiplication over addition, such as 3(2 + 5) = 3*2 + 3*5. It also gives an example of distributing subtraction over multiplication, such as rewriting (16 - 7)^3 using the distributive property.
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The document discusses the distributive property in mathematics. It defines the distributive property as the property that states if you divide both sides of an equation by the same nonzero number, the new equation will have the same solution. It provides examples of distributing multiplication over addition, such as 3(2 + 5) = 3*2 + 3*5. It also gives an example of distributing subtraction over multiplication, such as rewriting (16 - 7)^3 using the distributive property.
Integral Exponents
This document discusses integral exponents and how to evaluate expressions with zero and negative exponents. It provides examples of simplifying expressions with zero and negative exponents by using the definition that a negative exponent means to take the reciprocal of the base and raise it to the positive value of the exponent. It also explains that any nonzero number raised to the zero power is equal to 1, and expressions should be rewritten with only positive exponents.
Exponents.Test.Slide.Show
Exponents describe the number of times a base is used as a factor. The base is the number being multiplied and the exponent shows how many times to multiply the base. For example, 2 to the 12th power means multiplying 2 by itself 12 times, or 2 x 2 x 2 x... = 4,096. Exponents allow large products to be written concisely using the format of base^exponent.
Exponent properties
This document summarizes exponent properties for adding, subtracting, multiplying, and dividing terms with exponents. It provides examples for each property:
1) When adding or subtracting terms with the same base and exponent, only the coefficient changes.
2) When multiplying terms, you add the exponents and multiply the coefficients.
3) When dividing terms, you subtract the bottom exponent from the top and divide the coefficients.
4) When simplifying powers of powers, you distribute the exponent and multiply the exponents.
Multiplying polynomials
1) Squaring a binomial expression like (x + 3) means multiplying it by itself. This results in terms of x^2, x, and a constant term.
2) The coefficient of the middle (x) term is always 2 times the constant term. The constant term is the constant multiplied by itself.
3) Squaring expressions with subtraction, like (x - 4)^2, follows the same pattern but the middle term is negative.
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This document discusses how to simplify exponential expressions using a shortcut rather than explicitly multiplying out terms with the same base. It states that (3x2)3 can be written as 27x6 by using the rule that when the same base is raised to powers that are multiplied, the powers are added instead of multiplied out.
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This document discusses exponents and exponent rules. It defines exponents as abbreviations representing very large or small numbers as a power to which a number is raised. The exponent tells how many times the base is used in multiplication. Exponent rules include: adding exponents when multiplying the same base; subtracting exponents when dividing the same base; multiplying exponents when raising a power to a power; and raising each term to the power when a product is raised to a power. The document also covers zero and negative exponents.
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This document discusses antiderivatives and indefinite integration. It explains that an antiderivative is a function whose derivative is equal to a given function, and that the general solution to a differential equation involving antiderivatives contains an arbitrary constant of integration. It provides examples of finding antiderivatives using basic integration rules and rewriting integrands before integrating. The key points are that antiderivatives are defined up to an additive constant, and that rewriting the integrand is an important step in the integration process.
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This document discusses antiderivatives and indefinite integration. It explains that an antiderivative is a function whose derivative is equal to a given function, and that the general solution to a differential equation involving antiderivatives contains an arbitrary constant of integration. It provides examples of finding antiderivatives using basic integration rules and rewriting integrands before integrating. The key points are that antiderivatives are defined up to an additive constant, and that rewriting the integrand is an important step in the integration process.
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The document discusses various techniques for finding antiderivatives (indefinite integrals). It covers:
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3) Expanding expressions, like using FOIL, before taking the antiderivative term by term.
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The document discusses various techniques for evaluating indefinite integrals (antiderivatives), including:
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5) Setting up word problems involving integrals to find related functions like total cost, revenue, distance over time.
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This document discusses antiderivatives and indefinite integration. It explains that an antiderivative is a function whose derivative is equal to a given function, and that the general solution to a differential equation involving antiderivatives contains an arbitrary constant of integration. It provides examples of finding antiderivatives using basic integration rules and rewriting integrands before integrating. The key points are that antiderivatives are defined up to an additive constant, and that rewriting the integrand is an important step in the integration process.
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This document discusses antiderivatives and indefinite integration. It explains that an antiderivative is a function whose derivative is equal to a given function, and that the general solution to a differential equation involving antiderivatives contains an arbitrary constant of integration. It provides examples of finding antiderivatives using basic integration rules and rewriting integrands before integrating. The key points are that antiderivatives are defined up to an additive constant, and that rewriting the integrand is an important step in the integration process.
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1) The derivative of the exponential function ex is equal to ex.
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x^2(x-5)(x-6) = x^2(x-5)(x-6)
Canceling the common factors, we obtain:
x^2 = x^2
Since this holds for all real values of x, the solution is all real numbers. Therefore, the solution is x ∈ R.
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Exponent exercises
- 1. Warm-upSession 17 Simplify each expression completely. 83 92 34 336 x.4 48.3 45.2 .1 xx aab xx yx 918 yx 7 000,40 x 211 256 ba 12 x
- 2. Homework Check You can use your anything in your notebook to help you, just not your text book. Write each expression as a single power of the base. 1. 73 76 2. (y4)3 Simplify each expression. 3. (2y)4 4. (3b3)2 b Turn this in on the cart. Make sure your name is on your paper.
- 3. Zero & Negative Exponents
- 4. Negative exponents move terms to the other part of the fraction. (reciprocal) Example 1: Evaluate x y 2 4 x y2 4 4 is downstairs. We will move it… x y16
- 5. Example 2 4 3 x 1 3 4 x 4 3x x is downstairs. We will move it…
- 6. Example 3 2 9 2 9 1 81 1 -9 is upstairs. We will move it…
- 7. Now let’s include what we learned from last class. 33 44 33 4 0 4 1 42 3 8 3 6561 What do we know that we can do with these exponents? What do we know that we can do with these exponents? 2 25 2 10 2 )10( 1 100 1
- 8. Evaluate the expression without using a calculator. 2 21 32 43.6 3.5 44.4 - 4 1 9 144 1
- 9. Rewrite the expression with positive exponents. 3 2 33 5.9 3 .8 2.7 b x yx 3 3 2 x y 2 3x 3 125 1 b