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Chapter 6: Problem 393

Simplify. $$\left(\frac{y^{4}}{z^{10}}\right)^{5}$$

Short Answer

Expert verified
\frac{y^{20}}{z^{50}}

Step by step solution

01

Apply the Power Rule

Use the power rule for exponents, which states that \(\big(\frac{a}{b}\big)^m = \frac{a^m}{b^m}\), to distribute the exponent outside the fraction to both the numerator and the denominator. \(\big(\frac{y^4}{z^{10}}\big)^5 = \frac{(y^4)^5}{(z^{10})^5}\).
02

Simplify the Numerator

Raise the numerator to the power of 5. To do this, multiply the exponents: \(y^{4 \times 5} = y^{20}\).
03

Simplify the Denominator

Raise the denominator to the power of 5. Multiply the exponents: \(z^{10 \times 5} = z^{50}\).
04

Combine the Results

Combine the simplified numerator and denominator: \(\frac{y^{20}}{z^{50}}\).

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Key Concepts

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

Simplifying Fractions
Simplifying fractions is a fundamental concept in algebra. A fraction is composed of two parts: a numerator (top part) and a denominator (bottom part). To simplify a fraction, you must reduce it to its simplest form.
This can be done by dividing both the numerator and the denominator by their greatest common divisor (GCD). If the numerator and denominator are both powers, simplifying involves reducing each exponent by factoring out common bases.
Exponent Multiplication
Exponent multiplication occurs when you raise a power to another power. The power rule for exponent multiplication states that when you raise an exponent to another exponent, you multiply the exponents.
The general formula is \( (a^m)^n = a^{m \times n} \). In our exercise, we applied this rule separately to both the numerator and the denominator. For the numerator \( y^4 \), raised to the power of 5, you multiply 4 by 5: \( y^{4 \times 5} = y^{20} \). The same rule applies to the denominator.
Numerator and Denominator Simplification
In algebra, simplifying the numerator and the denominator individually aids in reducing the overall fraction.
Following the power rule, the numerator and denominator are raised to the given exponent. For example, \( (y^4)^5 = y^{20} \) and \( (z^{10})^5 = z^{50} \).
Once simplified separately, combine these results to form the simplified fraction: \( \frac{y^{20}}{z^{50}} \).
This approach keeps things organized and ensures accuracy in simplification.

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Most popular questions from this chapter

Simplify. (a) \(-5^{-3}\) (b) \(\left(-\frac{1}{5}\right)^{-3}\) |(c) \(-\left(\frac{1}{5}\right)^{-3}\) (d) \((-5)^{-3}\)

Divide \(\frac{10 x^{2}+x-12}{2 x}\) and explain with words how you get each term of the quotient.

Divide the monomials. $$\frac{18 a^{4} b^{8}}{-27 a^{9} b^{5}}$$

Divide each polynomial by the monomial. $$\left(8 x^{3}+6 x^{2}\right) \div 2 x$$

Mixed Practice $$\frac{63 r^{6} s^{3}}{9 r^{4} s^{2}}-\frac{72 r^{2} s^{2}}{6 s}$$
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