Problem 75 - Algebra 2

Question

Simplify each expression. Assume that all variables are positive. $$\left[\left(x^{-\frac{1}{2}}\right)^{2}\right]^{\frac{1}{3}}$$

Step-by-Step Solution

Answer

The simplified form of the expression \(\left[\left(x^{-\frac{1}{2}}\right)^{2}\right]^{\frac{1}{3}}\) is \(x^{-\frac{1}{3}}\).

Step 1: Solve the Inner Exponent

The base value is \(x^{-\frac{1}{2}}\) and it's raised to the power of 2. According to the exponent rule, we can multiply the powers. This will yield the term \((x^{-\frac{1}{2} \cdot 2})\), which simplifies to \(x^{-1}\).

Step 2: Solve the Outer Exponent

The result of our previous calculation \(x^{-1}\) is raised to the power of \(\frac{1}{3}\). Once again, we multiply the powers to receive our simplified expression, which results in \(x^{-1 \cdot \frac{1}{3}}\).

Step 3: Simplify the Expression

Simplify \(x^{-1 \cdot \frac{1}{3}}\) to derive the final simplified answer, \(x^{-\frac{1}{3}}\).

Key Concepts

Simplifying Expressions

Simplifying expressions is like tidying up a messy room. You arrange everything in the simplest form so it's easier to understand. When dealing with algebraic expressions that include exponents, your goal is to reduce them step by step, applying rules that help you climb to the answer.
To simplify expressions, follow these main steps:

• Identify parts of the expression you can simplify. Look for terms with the same base, like in the expression where we have repeated uses of the base \(x\).
• Apply rules for exponents to combine and reduce those parts sequentially.
• Ensure you're always expressing numbers and variables in their most compact form.

As you practice more, you'll get quicker at seeing which parts can be tackled first and how to efficiently simplify any expression.

Exponent Rules

Exponent rules are the guidelines we follow when working with expressions that involve powers. They allow us to break down and simplify complex problems. Here's a rundown of your foundational exponent rules, which you'll use repeatedly:

• **Product of Powers Rule**: When multiplying two expressions with the same base, add their exponents: \(a^m \times a^n = a^{m+n}\).
• **Power of a Power Rule**: When raising a power to another power, multiply the exponents: \((a^m)^n = a^{m\cdot n}\).
• **Power of a Product Rule**: When raising a product to an exponent, distribute the exponent: \((ab)^n = a^n \times b^n\).
• **Quotient of Powers Rule**: When dividing two expressions with the same base, subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
• **Zero Exponent Rule**: Any non-zero base raised to the power of zero is 1: \(a^0 = 1\).

In our given problem, the 'Power of a Power' rule was key, as we multiplied the exponents step by step to simplify \(x^{\left( -\frac{1}{2} \right) \times 2}\) and then \(x^{-1 \times \frac{1}{3}}\) with ease.

Negative Exponents

When it comes to negative exponents, they might seem tricky, but they're not too hard once you understand their meaning. A negative exponent indicates that you take the reciprocal of the base raised to the opposite positive power.
To put it simply:

• \(x^{-n}\) can be rewritten as \(\frac{1}{x^n}\)

So when you see \(x^{-1}\), flip it over to become \(\frac{1}{x}\).
In our solved exercise, you simplified \(x^{-1}\) directly, but remember, the same principle applies if you want to picture it as a fraction instead. It's handy to know this when combining terms or flipping them for easier understanding.Understanding negative exponents arms you with the ability to quickly rearrange and simplify expressions involving fractions and division connected to powers. Keep practicing, and these notions will become second nature!