Question

Write each expression without parentheses. Assume all variables are positive. $$ \left(\frac{a}{b}\right)^{5} $$

Short answer

Answer: \(\frac{a^5}{b^5}\)

Step-by-step solution

Identify the exponent outside the parentheses

In this case, we have an expression inside parentheses raised to the power of 5: \(\left(\frac{a}{b}\right)^{5}\).

Distribute the exponent to both terms inside the parentheses

Using the power property \( \left(\frac{a}{b}\right)^n = a^n/b^n\), we will distribute the exponent 5 to both terms (numerator and denominator) inside the parentheses: $$ \left(\frac{a}{b}\right)^{5} = \frac{a^5}{b^5}. $$

Write the final expression without parentheses

Now that we have distributed the exponent, we can write the expression without parentheses: $$ \frac{a^5}{b^5}. $$ This is the simplified expression without parentheses.

Key concepts

Expressions with Parentheses

Expressions with parentheses are a common way to group terms in algebra. Parentheses tell us which parts of the expression to evaluate first. For instance, in the expression \(\left(\frac{a}{b}\right)^{5}\), the fraction \(\frac{a}{b}\) is grouped together.

• The parentheses help identify the base of the expression when there is an exponent outside.
• It's like having a box around the fraction, stating that everything inside the parentheses should be addressed as a unit first before applying the power outside.

Recognizing the role of parentheses is essential for correctly simplifying expressions. This initial step of grouping helps prevent errors in calculations.

Power Property

The power property is a rule in algebra that tells us how to handle exponents. If we have an expression like \(\left(\frac{a}{b}\right)^n\), the power property tells us that we can distribute the exponent across the fraction.

• This means that \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\).
• This is a very helpful property, especially when simplifying expressions with fractions.

So in our initial expression, the exponent 5 applies to both the numerator \(a\) and the denominator \(b\). By applying the power property, we rewrite the expression as \(\frac{a^5}{b^5}\), maintaining the relationship between numerator and denominator.

Simplifying Expressions

Simplifying expressions is all about making them as straightforward as possible. It means reducing the expression to a form that is easier to understand and work with.

• In our example, we began with \(\left(\frac{a}{b}\right)^5\), a more complex form due to the parentheses and external exponent.
• After applying the power property, we simplified this to \(\frac{a^5}{b^5}\), a clearer and easier form.

When you simplify an expression, you make it more manageable for further mathematical operations or for solving equations. Always confirm the simplified form by comparing it back to your original terms to ensure correctness.