Problem 43 - Algebra 1

Question

Rewrite the expression with positive exponents $$\frac{8}{m^{-2}}$$

Step-by-Step Solution

Answer

The expression \( \frac{8}{m^{-2}} \) with positive exponents is \( 8 \cdot m^2 \).

Step 1: Identify the negative exponent

The term with the negative exponent is \( m^{-2} \) in the denominator.

Step 2: Apply the rule of negative exponent

By the rule of negative exponent, we rewrite \( m^{-2} \) as \( \frac{1}{m^2} \). So, the expression becomes \( \frac{8}{\frac{1}{m^2}} \).

Step 3: Simplify the expression

This is a division of fractions. To divide fractions, we multiply by the reciprocal of the divisor. Hence, we multiply 8 by \( m^2 \) to get the final answer of \( 8 \cdot m^2 \).

Key Concepts

Exponents

Exponents are a way to express repeated multiplication of a base number. If you see something like \(a^n\), it means that the base \(a\) is multiplied by itself \(n\) times. Here’s a quick breakdown:

• \(a^2\) (or "a squared") means \(a \cdot a\).
• \(a^3\) (or "a cubed") means \(a \cdot a \cdot a\).
• In general, \(a^n\) indicates the base \(a\) is used as a factor \(n\) times.

Exponents simplify multiplication problems and make it easier to handle large numbers and equations. Understanding exponents is fundamental in algebra and comes in handy in various math operations, such as working with polynomials or scientific notation.

Negative Exponents

Negative exponents can look confusing, but they follow a simple rule that makes them quite handy. The rule is that any non-zero number with a negative exponent can be rewritten as its reciprocal with a positive exponent. For example,

• \(b^{-n} = \frac{1}{b^n}\).

This means you are essentially "flipping" the base to the other side of the fraction line. If a term with a negative exponent is in the denominator, like \(m^{-2}\), it moves to the numerator as \(m^2\).
Understanding negative exponents is crucial since they allow us to express everything using positive exponents, which simplifies many equations and expressions in algebra.

Division of Fractions

When dividing fractions, the process involves multiplying by the reciprocal of the divisor.

• If you have a division expression like \(\frac{a}{\frac{b}{c}}\), you multiply \(a\) by the reciprocal of \(\frac{b}{c}\), which is \(\frac{c}{b}\).
• This results in \(a \cdot \frac{c}{b}\).

This trick makes division much more manageable and is a powerful tool in algebra.
With the exercise presented, dividing the expression \(\frac{8}{\frac{1}{m^2}}\) involves flipping the divisor \(\frac{1}{m^2}\) to become \(m^2\), and then multiplying. Therefore, \(8 \cdot m^2\) is the simplified expression after applying division of fractions.