Problem 53 - Intermediate Algebra

Question

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ \frac{\left(x^{3}\right)^{1 / 2}}{x^{7 / 2}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{1}{x^2} \).

Step 1: Apply Power of a Power Rule

The Power of a Power Rule states that \[(a^m)^n = a^{m imes n}.\]Apply this rule to the expression in the numerator: \[(x^3)^{1/2} = x^{3 \times 1/2} = x^{3/2}.\]

Step 2: Rewrite the Expression with Simplified Power

Replace the numerator with the expression obtained in Step 1. Now the expression is:\[\frac{x^{3/2}}{x^{7/2}}.\]

Step 3: Apply Quotient Rule for Exponents

The Quotient Rule for exponents states that\[\frac{a^m}{a^n} = a^{m-n}.\]Apply this rule to the simplified expression from Step 2:\[x^{3/2 - 7/2} = x^{-4/2} = x^{-2}.\]

Step 4: Rewrite with Positive Exponent

The negative exponent rule states that \[a^{-m} = \frac{1}{a^m}.\]Apply this rule to rewrite the expression with a positive exponent:\[x^{-2} = \frac{1}{x^2}.\]

Key Concepts

Power of a Power Rule

When working with exponents, the power of a power rule is a useful tool. This rule helps simplify expressions where an exponent is raised to another exponent. The rule is:

• If you have an expression \((a^m)^n\), you multiply the exponents: \(a^{m \times n}\).

In our exercise, we applied this rule to the term \((x^3)^{1/2}\). Multiplying the exponents, we get \(x^{3 \times 1/2} = x^{3/2}\).
This operation makes it easier to handle expressions by reducing the complexity of multiple exponents. Always remember, the goal is to simplify and combine powers effectively!

Quotient Rule for Exponents

The quotient rule for exponents is crucial when dividing like bases. It simplifies the expression by subtracting the exponents. This can be expressed as:

• For a division expression \(\frac{a^m}{a^n}\), you subtract the exponents: \(a^{m-n}\).

In our example, after applying the power of a power rule, we had \(\frac{x^{3/2}}{x^{7/2}}\). Using the quotient rule, we subtracted the exponents: \(x^{3/2 - 7/2} = x^{-(4/2)} = x^{-2}\).
This step simplifies the expression further and readies it for rewriting with positive exponents.

Negative Exponent Rule

Now that we have a negative exponent, it’s time to simplify it using the negative exponent rule. This rule states that a base with a negative exponent can be rewritten as a reciprocal with a positive exponent:

• For an expression \(a^{-m}\), rewrite it as \(\frac{1}{a^m}\).

In our problem, we had \(x^{-2}\). Applying the negative exponent rule, we transformed it into \(\frac{1}{x^2}\).
This step is essential because usually we prefer writing expressions with positive exponents, which makes them simpler to interpret and use.

Simplifying Expressions

Simplifying expressions with exponents involves methodically applying exponent rules. Here’s a brief checklist:

• Use the power of a power rule to simplify any base raised to an exponent that is further raised to another exponent.
• Apply the quotient rule for exponents when dividing like bases to condense multiple terms into a single base with one exponent.
• Convert negative exponents to positive by rewriting them as fractions.

Each of these rules works hand-in-hand to transform complex expressions into manageable forms. Simplification is the key to clarity, and by following these methods, you can achieve neat, efficient expressions as seen with \(\frac{1}{x^2}\) in our example.
Having these techniques at your fingertips will also make your work more organized and will help you tackle more advanced mathematics with confidence.