Problem 371 In the following exercises, simp... [FREE SOLUTION]

91影视

Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

$Math Studyset 91影视 Explanations$ Math

2016 Edition

Chapter 10: Problem 371

In the following exercises, simplify. $$\left(m^{-2}\right)^{-3}$$

Short Answer

Expert verified

The expression simplifies to $m^{6}$.

Step by step solution

Evaluate the exponent

Start by applying the power of a power property in exponents. This property states that $(a^m)^n = a^{m \times n}$. Apply this property to $(m^{-2})^{-3}$.

Multiply the exponents

According to the power of a power property, multiply the exponents together: $-2 \times -3$. When you multiply $-2$ and $-3$, you get \6\.

Simplify the expression

This simplifies the expression to $m^{6}$.

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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 rule in mathematics that simplifies the handling of nested exponents. This property states that when you have an exponent raised to another exponent, you can multiply the exponents together. The general formula for this property is $(a^m)^n = a^{m \times n}$.

For instance, if you have $(m^{-2})^{-3}$, you can apply the power of a power property.

Using the property, multiply the exponents $-2$ and $-3$ together. This means $(m^{-2})^{-3}$ becomes \m^{-2 \times -3} = m^{6}\.

Understanding this property allows you to simplify complex exponent expressions more easily and with fewer steps.

Multiplying Exponents

Multiplying exponents often involves using the power of a power property. When dealing with exponents, multiplication occurs when an exponent is raised to another exponent.

In our example $(m^{-2})^{-3}$, you must multiply the exponents $-2$ and $-3$.

Here are some steps to follow:

Identify the base and the exponents.
Multiply the exponents together: $-2 \times -3 = 6$.
Simplify the result: \m^6\.

When multiplying exponents, always ensure you handle negative signs correctly. Negatives can switch the direction of the exponents, making it crucial to multiply them accurately.

Negative Exponents

Negative exponents indicate that the base should be taken as the reciprocal. A negative exponent \a^{-n}\ is equivalent to \frac{1}{a^n}\.

For example, \[m^{-2} = \frac{1}{m^2}\]. But in the context of the power of a power property, you don't reciprocate until after applying the property.

In $(m^{-2})^{-3}$, the negative signs on the exponents influence the multiplication step. Multiply $-2$ and $-3$ to get \m^6\. The multiplication effectively cancels out the negative signs, turning them into a positive exponent.

Correct handling of negative exponents is essential as it helps transition complex expressions into simpler, more manageable forms.

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91影视

In the following exercises, simplify. $$\left(m^{-2}\right)^{-3}$$

Short Answer

Step by step solution

Evaluate the exponent

Multiply the exponents

Simplify the expression

Key Concepts

Power of a Power Property

Multiplying Exponents

Negative Exponents

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