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Chapter 6: Problem 380

Simplify. $$\frac{\left(y^{3}\right)^{4}}{y^{10}}$$

Short Answer

Expert verified
The simplified form is \(y^2\).

Step by step solution

01

Apply the Power Rule for Exponents

Use the power rule \(\big(a^m\big)^n = a^{m \times n}\) to simplify the numerator. Here, \(y^3\) is raised to the 4th power, so \(\big(y^3\big)^4 = y^{3 \times 4} = y^{12}\). This gives us: \(\frac{y^{12}}{y^{10}}\).
02

Apply the Quotient Rule for Exponents

Use the quotient rule \(\frac{a^m}{a^n} = a^{m-n}\) to simplify the fraction. Subtract the exponent in the denominator from the exponent in the numerator: \(\frac{y^{12}}{y^{10}} = y^{12 - 10} = y^2\).

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Key Concepts

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

Power Rule for Exponents
The Power Rule for Exponents is a fundamental concept in simplifying exponential expressions. It states that when you raise a power to another power, you multiply the exponents. For example, \( (a^m)^n = a^{m \times n} \).

Let's break this down using our example: \( \frac{(y^3)^4}{y^{10}} \). Here, we see \( y^3 \) raised to the 4th power. According to the Power Rule, we'll multiply the exponents: \( (y^3)^4 = y^{3 \times 4} = y^{12} \).
This gives us the simplified form: \( \frac{y^{12}}{y^{10}} \).
This rule is particularly useful for simplifying expressions where exponential terms are nested within each other.
Quotient Rule for Exponents
The Quotient Rule for Exponents simplifies expressions involving division of the same base with different exponents. The rule states: \( \frac{a^m}{a^n} = a^{m-n} \).

Applied to our example: \( \frac{y^{12}}{y^{10}} \), we subtract the exponent of the denominator from the exponent of the numerator. This results in: \( y^{12 - 10} = y^2 \).

By applying the Quotient Rule, we see that the division of exponential terms simplifies into a single exponential term with a reduced exponent. This is an essential concept for managing large and complex exponent expressions.
Simplification of Fractions
Simplification of fractions introduces clarity and simplicity in mathematical expressions. When dealing with fractions involving exponents, the Power and Quotient rules are crucial tools.

Consider our example again: \( \frac{(y^3)^4}{y^{10}} \). Using the Power Rule, we first simplify the numerator: \( (y^3)^4 = y^{12} \). Next, using the Quotient Rule, we simplify the fraction: \( \frac{y^{12}}{y^{10}} = y^{12 - 10} = y^2 \).
This transformation turns a more complicated-looking fraction into a much simpler form: \( y^2 \).

Mastering these fraction simplification techniques is essential for tackling more complex algebraic problems and making math easier to handle.

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Most popular questions from this chapter

Mixed Practice $$\frac{27 a^{7}}{3 a^{3}}+\frac{54 a^{9}}{9 a^{5}}$$

Divide each polynomial by the monomial. $$\left(48 y^{4}-24 y^{3}\right) \div\left(-8 y^{2}\right)$$

Divide the monomials. $$\frac{\left(4 u^{2} v^{5}\right)\left(15 u^{3} v\right)}{\left(12 u^{3} v\right)\left(u^{4} v\right)}$$

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

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