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Chapter 10: Problem 254

In the following exercises, simplify. $$\frac{\left(y^{4}\right)^{3}}{y^{7}}$$

Short Answer

Expert verified
\(y^{5}\)

Step by step solution

01

Apply the Power of a Power Rule

When a power is raised to another power, multiply the exponents. So, \(\frac{\big(y^{4}\big)^{3}}{y^{7}}\rightarrow\frac{y^{4 \times 3}}{y^{7}}\). This simplifies to \(\frac{y^{12}}{y^{7}}\).
02

Subtract the Exponents

To divide like bases, subtract the exponent in the denominator from the exponent in the numerator: \(\frac{y^{12}}{y^{7}}\rightarrow y^{12-7}\). This gives \y^{5}\.
03

Simplified Expression

The simplified expression is \(y^{5}\).

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Key Concepts

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

Power of a Power Rule
One of the essential rules for working with exponents is the power of a power rule. This rule is used when you have an exponent raised to another exponent, as in \(\big(a^m\big)^n\).To simplify this, you multiply the exponents together.

Here's a quick breakdown:
  • Given \(\big(y^4\big)^3\), you multiply the exponents (4 and 3).
  • So, \(\big(y^4\big)^3 = y^{4 \times 3} = y^{12}\).
This step condenses a multi-layered exponential expression into a simpler one with a single exponent. Remember, this rule helps you simplify complex exponential expressions, cutting down on steps and possible errors.
Subtracting Exponents
Next comes another crucial concept: subtracting exponents. This rule applies when you are dividing like bases. If you have \(\frac{a^m}{a^n}\), you subtract the exponent in the denominator from the exponent in the numerator. This can make things much easier!

Take the simplified expression we got from the power of a power rule, \(\frac{y^{12}}{y^7}\):
  • Here, because the bases are the same (both are y), you subtract the exponent in the denominator (7) from the exponent in the numerator (12).
  • This gives \(\frac{y^{12}}{y^7} = y^{12-7} = y^5\).
Remember, getting the exponents correctly subtracted is a key step to ensuring your final expression is accurately simplified.
Simplified Expression
After applying the power of a power rule and subtracting the exponents, we arrive at the simplified expression. This is our goal: to take a complex-looking expression and make it as straightforward as possible.

From our example, \( \frac{(y^4)^3}{y^7} \), we've simplified it step-by-step to become \ y^5 \.This means:
  • Initially, after applying the power of a power rule, we got \ y^{12} \.
  • Then, after subtracting the exponent in the denominator from the exponent in the numerator, we got \ y^5 \.
This final, simplified expression is much easier to work with in further mathematical operations. Always aim to simplify expressions to their cleanest form for straightforward calculations and interpretations.

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