Problem 94 - College Algebra with Modeling and Visualization

Question

Simplify the expression and write it with rational exponents. Assume that all variables are positive. $$ \left(x^{-3 / 2}\right)^{2 / 3} $$

Step-by-Step Solution

Answer

The simplified expression is \(x^{-1}\).

Step 1: Understand the expression

We need to simplify the expression \( \left(x^{-3 / 2}\right)^{2 / 3} \) and express it using rational exponents.

Step 2: Apply the power of a power rule

The power of a power rule states that \( (a^m)^n = a^{m \cdot n} \). We apply this rule to the expression \( \left(x^{-3/2}\right)^{2/3} \).

Step 3: Multiply the exponents

Multiply the exponents: \(-\frac{3}{2} \times \frac{2}{3}\). This will give us the new exponent when the powers are combined.

Step 4: Simplify the multiplication

Calculate \(-\frac{3}{2} \times \frac{2}{3} = -1\).

Step 5: Rewrite the expression

Insert the calculated exponent back into the expression to get \(x^{-1}\).

Key Concepts

Power of a Power Rule

When dealing with exponents, especially when they are stacked, it's essential to understand the power of a power rule. This rule helps simplify complex exponentiated expressions.
Simply put, the power of a power rule states that if you have an expression like \((a^m)^n\), you can multiply the exponents together. So it transforms into \(a^{m \cdot n}\).

• This rule applies to both whole and rational exponents.
• It simplifies the expression without changing its value.

In our exercise, this rule allows us to tackle the expression \(\left(x^{-3/2}\right)^{2/3}\) by multiplying the exponents \(-3/2\) and \(2/3\) together.
Using this rule keeps things systematic and allows us to understand how to handle even more complicated expressions with ease.

Exponent Multiplication

Exponent multiplication is an essential concept in simplifying expressions with exponents. Here, we take the exponents coming from the power of a power rule and multiply them.
This operation follows basic fraction multiplication, even when dealing with negative or fractional numbers.

• In our given expression, we multiply \(-\frac{3}{2} \times \frac{2}{3}\).
• To multiply fractions, multiply the numerators together and the denominators together.
• So, \(-3 \times 2 = -6\) and \(2 \times 3 = 6\), combining to \(-\frac{6}{6}\).
• Simplify \(-\frac{6}{6}\) to get \(-1\).

Thus, the multiplication of the exponents results in \(-1\). This step is crucial to simplify the entire expression.

Expression Simplification

Once you have determined the new exponent from multiplying, it's time to rewrite the expression in its simplest form. Simplifying an expression makes it clearer and easier to understand or use in other calculations.
With our calculated exponent of \(-1\), the original expression \(\left(x^{-3/2}\right)^{2/3}\) simplifies to \(x^{-1}\).

• Write the expression with new exponents: simply \(x^{-1}\).
• This form is often more convenient for interpretation and further mathematical operation.

Interestingly, writing it as \(x^{-1}\) means you can easily convert it back to a fraction: \(\frac{1}{x}\) if needed.
Understanding each simplification step is fundamental, allowing for seamless transitions between different forms of mathematical expressions.