Problem 13 - Algebra for College Students

Question

Simplify each numerical expression. \(\frac{1}{\left(\frac{3}{7}\right)^{-2}}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{9}{49}\).

Step 1: Understand the Expression

The expression is a fraction involving an exponent: \(\frac{1}{\left(\frac{3}{7}\right)^{-2}}\). We have a fraction raised to a negative exponent.

Step 2: Apply the Negative Exponent Rule

The negative exponent rule states that \( a^{-n} = \frac{1}{a^n} \). In our case, \( \left(\frac{3}{7}\right)^{-2} = \frac{1}{\left(\frac{3}{7}\right)^2} \). Therefore, \( \frac{1}{\left(\frac{3}{7}\right)^{-2}} = \left(\frac{3}{7}\right)^2 \).

Step 3: Simplify the Fraction Exponent

Now we need to simplify \(\left(\frac{3}{7}\right)^2\). To do this, we square the numerator and the denominator separately: \(\left(\frac{3}{7}\right)^2 = \frac{3^2}{7^2} = \frac{9}{49}\).

Key Concepts

Negative Exponents

Negative exponents can seem tricky at first, but they follow a simple rule. When you have a negative exponent, it tells you to take the reciprocal of the base and switch the sign of the exponent to positive.
For example, if you have a base of \( a \) with an exponent of \(-n\), it converts to its reciprocal with a positive exponent: \( a^{-n} = \frac{1}{a^n} \).
This means you're flipping the base "upside down," if you think in terms of fractions. For expressions like \( \left(\frac{3}{7}\right)^{-2} \), you apply this rule by flipping the fraction and changing the exponent to positive, transforming it into \( \frac{1}{\left(\frac{3}{7}\right)^2} \).
This rule helps make calculations with negative exponents straightforward once you get the hang of it!

Fraction Operations

Understanding fractions is key to dealing with expressions involving them, especially when exponents are involved. Operations with fractions need a few simple steps:

• For multiplication, multiply the numerators together and the denominators together.
• For division, multiply by the reciprocal of the second fraction.
• For addition and subtraction, make sure the fractions have a common denominator.

In our exercise, we deal with raising a fraction to an exponent. This means that both the numerator and the denominator will be raised to the same power.
So, to simplify \( \left(\frac{3}{7}\right)^2 \), square both 3 and 7 individually, resulting in \( \frac{3^2}{7^2} = \frac{9}{49} \).
Remember, each part of the fraction is handled separately when processed through the exponent operation.

Simplifying Expressions

Simplifying expressions makes them easier to handle and interpret. The goal is to reduce them to their simplest form while maintaining their value.
In algebra, expressions are made up of numbers, variables, and operations. When you simplify, you are combining like terms and applying mathematical principles to make the expression more manageable.
For expressions with exponents, like \( \frac{1}{\left(\frac{3}{7}\right)^{-2}} \), simplifying involves applying exponent rules and fraction arithmetic to get to a straightforward result.
The process involved flipping the fraction due to the negative exponent (which leads to \( \left(\frac{3}{7}\right)^2 \)) and then simplifying that fraction to evaluate the expression. This results in the final simplified form: \( \frac{9}{49} \).
Understanding how to simplify not only helps solve the immediate problem but builds the ability to approach more complex problems with confidence.