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Questions and Answers
When raising a number to a power and then raising it to another power, you add the exponents together.
When raising a number to a power and then raising it to another power, you add the exponents together.
False
In the quotient of powers rule, when two identical bases are divided by each other, the exponents are added together.
In the quotient of powers rule, when two identical bases are divided by each other, the exponents are added together.
False
According to the zero exponent rule, any nonzero number raised to the power of 0 is equal to 0.
According to the zero exponent rule, any nonzero number raised to the power of 0 is equal to 0.
False
If we have 5^3 divided by 5^2, the result is 5.
If we have 5^3 divided by 5^2, the result is 5.
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Raising a nonzero number to a negative integer exponent results in a negative value.
Raising a nonzero number to a negative integer exponent results in a negative value.
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The result of 3^2 raised to the power of 4 is 3^8.
The result of 3^2 raised to the power of 4 is 3^8.
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In the power of a power rule, when a base raised to a power is also raised to another power, the two powers are added together.
In the power of a power rule, when a base raised to a power is also raised to another power, the two powers are added together.
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The power of a product rule states that when the product of two variables or numbers is raised to a power, the power is distributed evenly across each factor.
The power of a product rule states that when the product of two variables or numbers is raised to a power, the power is distributed evenly across each factor.
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According to the zero exponent rule, any base raised to the power of 0 equals 1.
According to the zero exponent rule, any base raised to the power of 0 equals 1.
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If we have 5^3 raised to the power of 2, the result would be 5^6.
If we have 5^3 raised to the power of 2, the result would be 5^6.
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In the power of a quotient rule, when a fraction is raised to an exponent, each part of the fraction is raised to that exponent separately.
In the power of a quotient rule, when a fraction is raised to an exponent, each part of the fraction is raised to that exponent separately.
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(2 * 3)^4 is equivalent to 2^4 * 3^4 according to the power of a product rule.
(2 * 3)^4 is equivalent to 2^4 * 3^4 according to the power of a product rule.
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Study Notes
Exponents and Powers
Introduction
Exponents and powers are fundamental mathematical concepts that describe repeated operations of arithmetic operations on a base value. These concepts are essential in various fields, including physics, engineering, finance, and computer science. Understanding these concepts allows us to perform complex calculations efficiently and accurately.
In this article, we will discuss the basic rules of exponents and powers, focusing on the subtopics of power of a power, power of a product, power of a number raised to an exponent, power of a quotient, and the zero exponent rule.
Power of a Power
The power of a power rule dictates that when a base raised to a power is also raised to another power, the two powers are multiplied, and the base remains the same. Mathematically, this can be represented as: (base^power)^new_power = base^(power × new_power).
For example, if we have 3^2 raised to the power of 3, we first calculate 3^2 as 9, then multiply the two powers to get 9^3, which equals 729. The result is consistent with the rule because (3^2)^3 = 3^(2 × 3) = 9^3.
Power of a Product
The power of a product rule states that when the product of two variables or numbers is raised to a power, the power is distributed evenly across each factor. Mathematically, this can be represented as: (a * b)^power = a^power * b^power.
For example, if we have (a + b) raised to the power of 3, the result is: (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3.
Power of a Number Raised to an Exponent
The rule for raising a number raised to an exponent to another power is given by: (base^exponent)^new_power = base^(exponent × new_power).
This means that when raising a base to an exponent and then raising it to another power, you multiply the exponents together and keep the base constant. For example, if we have 2^3 raised to the power of 2, the result is 2^(3 × 2) = 2^6, which equals 64.
Power of a Quotient
The quotient of powers rule states that when two identical bases are divided by each other, raised to some power, the exponents are subtracted from each other, and the base remains the same. Mathematically, this can be represented as: (base^power) / (base^new_power) = base^(power - new_power).
For example, if we have 4^3 divided by 4^2, we get: 4^3 / 4^2 = 4^(3 - 2) = 4^1, which equals 4.
Zero Exponent Rule
According to the zero exponent rule, any nonzero number raised to the power of 0 is equal to 1, while any nonzero number raised to the power of a negative integer is equivalent to its reciprocal. Mathematically, for n ≠ 0, we have: n^0 = 1 and n^-m = 1/n^m.
This means that dividing a number by itself repeatedly to the power of a positive integer will eventually reach 1, whereas raising it to a negative integer will make it approach its reciprocal.
In conclusion, understanding the rules of exponents and powers is crucial for performing complex mathematical operations. These rules provide valuable shortcuts and simplifications for calculating products, quotients, and raising expressions to different powers.
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Description
Test your understanding of the rules for exponents and powers with this quiz covering topics such as power of a power, power of a product, power of a number raised to an exponent, quotient of powers, and zero exponent rule. Improve your mathematical skills by practicing these fundamental concepts.
