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Chapter 10: Problem 258

In the following exercises, simplify. $$\frac{p^{11}}{\left(p^{5}\right)^{3}}$$

Short Answer

Expert verified
The simplified form is \(p^{-4}\).

Step by step solution

01

Simplify the exponent in the denominator

Rewrite the expression in the denominator by applying the power of a power property. For any integers a, m, and n: ewline \((a^m)^n = a^{m \times n}\)ewline Applying this property to \((p^5)^3\), we get: ewline\((p^5)^3 = p^{5 \times 3} = p^{15}\)
02

Write the new expression

Substitute the simplified denominator back into the original expression:ewline$$\frac{p^{11}}{p^{15}}$$
03

Apply the quotient rule for exponents

Use the quotient rule for exponents which states \(\frac{a^m}{a^n} = a^{m-n}\) where a is a nonzero number and m and n are integers.ewline Applying this to our expression, we get: ewline\(\frac{p^{11}}{p^{15}} = p^{11 - 15} = p^{-4}\)
04

Write the final simplified expression

The final simplified form of the expression is:ewline$$p^{-4}$$

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Key Concepts

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

Power of a Power Property
The power of a power property is a very useful rule in algebra, especially when dealing with exponents.
This rule states that when you have an exponentiated term raised to another exponent, you can simplify by multiplying the exponents together.
For example \[ (a^m)^n = a^{m \times n} \].
Let's apply this property to the given exercise. In the expression \((p^5)^3\), you can see that the base is \(p\) and the exponents are \(5\) and \(3\).
According to the property, we multiply these exponents together: \((p^5)^3 = p^{5 \times 3} = p^{15}\). Now, our expression simplifies to \(\frac{p^{11}}{p^{15}}\).
Quotient Rule for Exponents
The quotient rule for exponents helps you simplify expressions where the same base appears in both the numerator and the denominator.
This rule states \( \frac{a^m}{a^n} = a^{m - n} \) where \(a\) is a nonzero number, and \(m\) and \(n\) are integers.
In our exercise, we have \( \frac{p^{11}}{p^{15}} \).
Applying the quotient rule here, we subtract the exponent in the denominator from the exponent in the numerator:
\( p^{11 - 15} = p^{-4} \).
Thus, the simplified form of our expression so far is \( p^{-4} \).
Negative Exponents
Negative exponents might seem tricky, but they are easy to understand once you grasp the concept.
A negative exponent indicates the reciprocal of the base raised to the opposite positive exponent. That is, \( a^{-n} = \frac{1}{a^n} \).
In our exercise, we simplified the expression to \( p^{-4} \).
To convert this to a more standard form, we use the negative exponent rule: \( p^{-4} = \frac{1}{p^4} \).
So the final simplified form of our original expression \( \frac{p^{11}}{(p^5)^3} \) is \( \frac{1}{p^4} \).

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