Question

If \(\left(x^{2}\right)^{\frac{1}{5}}+\sqrt[5]{32 x^{2}}=a x^{\frac{2}{5}} \quad\) for all values of \(x\), what is the value of a? (A) 0 (B) 3 (C) 5 (D) 16

Short answer

The value of \(a\) is 3.

Step-by-step solution

Simplify the first term

We start by simplifying the term \((x^2)^{\frac{1}{5}}\). Applying the power to a power rule \((a^m)^n = a^{m \cdot n}\), we have \(x^{2 \cdot \frac{1}{5}} = x^{\frac{2}{5}}\).

Simplify the second term

Next, simplify the term \(\sqrt[5]{32x^2}\) which is the same as \((32x^2)^{\frac{1}{5}}\). We can separate the terms inside the radical: \(32^{\frac{1}{5}} \cdot (x^2)^{\frac{1}{5}}\). \(32 = 2^5\), so \(32^{\frac{1}{5}} = 2\). This results in \(2 \cdot x^{\frac{2}{5}}\).

Combine simplified terms

Add the simplified terms from Step 1 and Step 2: \(x^{\frac{2}{5}} + 2 \cdot x^{\frac{2}{5}}\) can be rewritten as \(3x^{\frac{2}{5}}\).

Equate to given expression

The original equation was given as \(a \cdot x^{\frac{2}{5}}\). From the previous step, we know the expression simplifies to \(3x^{\frac{2}{5}}\). Equating coefficients of like terms, we have \(a = 3\).

Key concepts

Powers and Exponents

Powers and exponents form the backbone of many algebraic expressions.
When dealing with expressions involving powers, you have to remember certain rules like the power to a power rule.
This states that \((a^m)^n = a^{m \cdot n}\). Essentially, this means that you multiply the exponents together. In our exercise, when you have \((x^2)^{\frac{1}{5}}\), you multiply 2 and \(\frac{1}{5}\) giving you \(x^{\frac{2}{5}}\).

Understanding how exponents behave and how to manipulate them correctly is vital.
This simplification is quite common in solving equations where power terms need to be handled first.
Knowing such rules will help you in simplifying expressions swiftly and accurately.
Keep practicing these rules to master handling complex expressions involving powers and exponents.

Radicals

Radicals are another important concept in algebra that involves roots of numbers or expressions.
When you see a radical expression like \(\sqrt[5]{32x^2}\), it's an indication that you are dealing with the 5th root.
In simpler terms, this means you need to find the value that when raised to the 5th power gives you the value under the radical.

A useful strategy with radicals is to express the numbers involved as powers.
In our example, since 32 can be written as \(2^5\), you can simplify \(32^{\frac{1}{5}}\) to 2 because you are essentially undoing the power.
This technique greatly helps in simplifying the expressions. Remember, breaking down complex expressions systematically is key when working with radicals.

Practicing how to simplify radicals will make it easier to recognize these transformations and implement them in solving algebraic expressions and equations.

Equations

Solving equations often involves combining like terms and making both sides of the equation equal through simplification.
This means breaking the expression down into more manageable parts.
In the original exercise, you see this when the simplified terms \(x^{\frac{2}{5}}\) and \(2 \, x^{\frac{2}{5}}\) are combined to form a single term \(3x^{\frac{2}{5}}\).

It's fundamental to understand how to make each side of the equation equal.

• Start by simplifying individual terms which often involves using knowledge from powers and radicals.
• Combine like terms so that the equation is easier to solve.
• Finally, equate both sides to find the variable or constant you're solving for, in this case, the coefficient \(a\).

In our exercise, both sides had \(x^{\frac{2}{5}}\), making it simple to equate coefficients, and thus determine \(a = 3\).
Understanding how to manipulate equations efficiently is crucial for accurately and quickly finding solutions.