Question

Simplify each expression. Assume that all variables are positive when they appear. $$(\sqrt[3]{3} \sqrt{10})^{4}$$

Short answer

3^{4/3} \cdot 100

Step-by-step solution

Understand the Problem

The given expression is \(\left(\sqrt[3]{3} \sqrt{10}\right)^4\). We need to simplify this expression while assuming all variables are positive.

Simplify the Radicals

Recall the properties of radicals: \(\sqrt[3]{a} = a^{1/3}\) and \(\sqrt{b} = b^{1/2}\). Substitute these into the expression: \(\left(3^{1/3} \cdot 10^{1/2}\right)^4\).

Apply Power Rule for Exponents

Use the power rule \((a^m \cdot b^n)^p = a^{m\cdot p} \cdot b^{n\cdot p}\). Applying this property to the expression yields: \(\left(3^{1/3} \cdot 10^{1/2}\right)^4 = 3^{1/3 \cdot 4} \cdot 10^{1/2 \cdot 4} = 3^{4/3} \cdot 10^2\).

Combine the Exponents

The expression can now be written as the product of two simplified terms: \(3^{4/3} \cdot 100\).

Key concepts

radicals

Radicals are expressions that contain a root symbol (\root ). They are a way to describe numbers that are not perfect squares or cubes. The two most common types of radicals are square roots, \(\root{a}\), and cube roots, \(\root[3]{b}\).

When simplifying radicals:

• Convert the radical to an exponent.
• Use the property: \(\root[n]{a} = a^{1/n}\).

Applying this to our problem: \(\sqrt[3]{3}\sqrt{10})^{4}\) becomes \(3^{1/3} \cdot 10^{1/2})^4\).

It's now easier to handle and prepare for further simplification.

properties of exponents

Exponents are shorthand for repeated multiplication of the same number. The main properties to remember are:

• Product of powers: \(a^m \cdot a^n = a^{m+n}\)
• Power of a power: \( (a^m)^n = a^{m \cdot n}\)
• Power of a product: \( (a \cdot b)^m = a^m \cdot b^m\)

In our exercise, we use the power of a product rule. So, \( (3^{1/3} \cdot 10^{1/2})^4 \) becomes \(3^{(1/3 \cdot 4)} \cdot 10^{(1/2 \cdot 4)}\).

Mathematically, this translates to \(3^{4/3} \cdot 10^2\). Now, each term is easier to combine and simplify.

simplifying expressions

Simplifying expressions involves using algebraic rules to make an expression more compact or easier to work with.

For the given problem, after applying the properties of radicals and exponents, you get \(3^{4/3} \cdot 100\).

When simplifying:

• First, handle any exponents.
• Second, multiply the simplified terms.
• Simplify any constants if possible.

Our final simplified form is \(3^{4/3} \cdot 100\). This is a neat and compact way to present the expression.