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Chapter 8: Problem 92

Simplify each expression. $$\frac{\left(m^{2}\right)^{4} m^{-1}}{\left(m^{3}\right)^{-1}}$$

Short Answer

Expert verified
The simplified expression is \( m^{10} \).

Step by step solution

01

Simplify the numerator

Start with the numerator: \ \( (m^2)^4 \) and \( m^{-1} \). When raising a power to another power, multiply the exponents. So, \( (m^2)^4 = m^{(2 \times 4)} = m^8 \). Therefore, the numerator becomes \( m^8 \times m^{-1} \).
02

Combine the exponents in the numerator

Combine the exponents in the numerator using the property \( a^m \times a^n = a^{m+n} \). So, \( m^8 \times m^{-1} = m^{8-1} = m^7 \).
03

Simplify the denominator

Now, simplify the denominator \ \( (m^3)^{-1} \). When an exponent is negative, it indicates the reciprocal. Thus, \( (m^3)^{-1} = \frac{1}{m^3} \).
04

Divide the simplified numerator by the denominator

Divide the simplified numerator \( m^7 \) by the simplified denominator \( \frac{1}{m^3} \). This is the same as multiplying by the reciprocal: \( m^7 \times m^3 = m^{7+3} = m^{10} \).

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Key Concepts

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

Exponent Rules
Understanding exponent rules is crucial for algebraic simplification.
First, you need to know how to handle powers and bases. Here are some key rules:
  • Product Rule: When you multiply two powers with the same base, you add their exponents. So, if you have \(a^m \times a^n\), it becomes \(a^{m+n}\).

  • Power of a Power Rule: When raising a power to another power, you multiply the exponents. For example, \((a^m)^n = a^{m \cdot n}\).

  • Quotient Rule: When you divide two powers with the same base, you subtract their exponents. So, \(\frac{a^m}{a^n} = a^{m-n}\).

  • Negative Exponent Rule: A negative exponent means you take the reciprocal of the base. So, \(a^{-m} = \frac{1}{a^m}\).
Using these rules helps simplify complex expressions, like the one in our exercise.
Reciprocal
The concept of reciprocal is tied to negative exponents. When you see a negative exponent, think of it as flipping the base upside down.
For instance, if you have \(m^{-1}\), this is the same as \(\frac{1}{m}\). Also, \((m^3)^{-1}\) turns into \(\frac{1}{m^3}\).

To simplify, always remember: negative exponent = reciprocal. This is a powerful tool to simplify expressions and make calculations easier. Especially when dividing terms with exponents.
Combining Exponents
Combining exponents properly simplifies algebraic expressions effectively. Here's how:
  • Addition of Exponents: According to the product rule, you add exponents when multiplying similar bases. For example, in the numerator of our exercise, we have \(m^8\) and \(m^{-1}\). Combining them gives us \(m^8 \times m^{-1} = m^{8-1} = m^7\).

  • Subtraction of Exponents: When dividing, you subtract the exponents. Reciprocals play a big role here. The denominator \((m^3)^{-1}\) simplifies to \(\frac{1}{m^3}\).
    So dividing by \(\frac{1}{m^3}\) is the same as multiplying by \(m^3\), which follows the rule of combining exponents: \( m^7 \times m^3 = m^{7+3} = m^{10}\).

Combining exponents helps handle larger expressions quickly, making algebra more manageable.

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Most popular questions from this chapter

Simplify each radical. Assume that all variables represent nonnegative real numbers. $$ \sqrt{36 z^{2}} $$

Simplify each radical. $$ \sqrt[3]{\frac{64}{125}} $$

Solve each equation. $$ x(x+2)=3 $$

Solve each equation. (Hint: In Exercises 67 and 68, extend the concepts to fourth root radicals.) $$ \sqrt[3]{5 x^{2}-6 x+2}=\sqrt[3]{x} $$

Work Exercises, in order, to see why a common student error is indeed an error. Use the distributive property to write \(6(5+3 x)\) as a sum.
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