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Chapter 1: Problem 7

State whether the equation is an example of either the product rule, the quotient rule, the power rule, raising a product to a power, or raising a quotient to a power. $$ \left(\frac{a}{4}\right)^{7}=\frac{a^{7}}{4^{7}} $$

Short Answer

Expert verified
Raising a quotient to a power.

Step by step solution

01

- Identify the type of equation

Look at the equation \(\left(\frac{a}{4}\right)^{7}=\frac{a^{7}}{4^{7}}\). Notice that it involves raising a fraction to a power.
02

- Recall the rule for raising a quotient to a power

The rule for raising a quotient to a power states that \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\). This means each part of the fraction is raised to the power separately.
03

- Confirm the rule matches the equation

Compare the given equation with the general form \(\left(\frac{a}{b}\right)^n\). We have \(\left(\frac{a}{4}\right)^{7}\) which matches the pattern, confirming it is an example of raising a quotient to a power.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponent Rules
Exponents are a way to express repeated multiplication of the same number. There are several rules for working with exponents, which help simplify expressions and solve equations:
  • Product Rule: When multiplying two expressions with the same base, add the exponents: ewline \(a^m \times a^n = a^{m+n}\).

  • Quotient Rule: When dividing two expressions with the same base, subtract the exponents: ewline \(\frac{a^m}{a^n} = a^{m-n}\).

  • Power Rule: When raising an exponent to another power, multiply the exponents: ewline \((a^m)^n = a^{mn}\).

  • Raising a Product to a Power: When raising a product to a power, raise each factor to the power: ewline \((ab)^n = a^n b^n\).

  • Raising a Quotient to a Power: When raising a quotient to a power, raise both the numerator and denominator to the power: ewline \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\).
Understanding these rules is essential for simplifying expressions and solving equations that involve exponents.
Raising a Fraction to a Power
When handling fractions in exponents, it's important to know how to raise both the numerator and the denominator to the given power. This involves:
  • Numerator: Raise the number in the numerator to the power: \((a)^n = a^n\).
  • Denominator: Raise the number in the denominator to the power: \((b)^n = b^n\).

    • Using the rule for raising a quotient to a power,
    the fraction \(\left(\frac{a}{b}\right)^n\) becomes \(\frac{a^n}{b^n}\). This means you apply the exponent to both the top and bottom separately.
    For example, with our specific case, \(\left(\frac{a}{4}\right)^7\), you raise both 'a' and '4' to the power of 7:
    ewline \(\left(\frac{a}{4}\right)^7 = \frac{a^7}{4^7}\).
Quotient Rule for Exponents
The quotient rule for exponents simplifies the division of two exponential expressions with the same base. The key principle of this rule is:
  • Subtract the exponent in the denominator from the exponent in the numerator: ewline \(\frac{a^m}{a^n} = a^{m-n}\).
This rule is especially helpful when dealing with fractions.
For example, if we have \(\frac{x^5}{x^2}\), this simplifies to: ewline \(x^{5-2} = x^3\).
Remember, the bases must be the same for the quotient rule to apply. In this case, since 'a' in \(\frac{a}{4}\) is being raised to the same power in both the numerator and the denominator, the quotient rule gives us a straightforward way to simplify the expression once the power has been applied separately.

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Most popular questions from this chapter

Calculate using the rules for order of operations. If an expression is undefined, state this. $$ -(-3)^{2} $$

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