Problem 78 - Algebra 2

Question

Simplify each expression. \(\left(b^{\sqrt{6}}\right)^{\sqrt{24}}\)

Step-by-Step Solution

Answer

The expression simplifies to \( b^{12} \).

Step 1: Understand the Power of a Power Rule

The expression \( \left(b^{\sqrt{6}}\right)^{\sqrt{24}} \) contains a power raised to another power. We use the rule \( \left(a^m\right)^n = a^{m \cdot n} \) to simplify such expressions.

Step 2: Identify the Exponents

Here, the base is \( b \), and the exponents are \( m = \sqrt{6} \) and \( n = \sqrt{24} \). We need to find the product of these two exponents to simplify the expression.

Step 3: Calculate the Exponent Product

The exponent in the simplified expression is \( m \cdot n = \sqrt{6} \times \sqrt{24} \). Use the property \( \sqrt{a} \times \sqrt{b} = \sqrt{a \cdot b} \) to simplify: \( \sqrt{6} \times \sqrt{24} = \sqrt{6 \times 24} \).

Step 4: Simplify Under the Radical

Calculate \( 6 \times 24 = 144 \). Thus, \( \sqrt{6 \times 24} = \sqrt{144} \).

Step 5: Simplify the Square Root

The square root of 144 is \( \sqrt{144} = 12 \). Thus, the expression simplifies to \( b^{12} \).

Key Concepts

Understanding the Power of a Power Rule

When you have an expression like \( (a^m)^n \), it means that a base, \( a \), is raised to an exponent \( m \), and then that result is raised to another exponent \( n \). The power of a power rule helps us simplify this by multiplying the exponents together. So, \( (a^m)^n \) becomes \( a^{m \cdot n} \). This rule is very useful in making complex expressions simpler and easier to work with. In our case, we have \( \left(b^{\sqrt{6}}\right)^{\sqrt{24}} \), which simplifies to \( b^{\sqrt{6} \cdot \sqrt{24}} \).

Simplifying Radicals

Radicals can seem tricky at first, but breaking them down can make them manageable. A radical, such as \( \sqrt{a} \), is the opposite of squaring a number. When simplifying radicals, one useful trick is using the property \( \sqrt{a} \times \sqrt{b} = \sqrt{a \cdot b} \). By multiplying the numbers under the radical, we can combine radicals into a single simpler form. For example, in our expression, \( \sqrt{6} \times \sqrt{24} \) simplifies to \( \sqrt{6 \times 24} \) by this rule, helping us further along our simplification journey.

Working with Exponent Products

Exponents represent repeated multiplication, so when multiplying bases with exponents, we need to consider how to handle the exponents efficiently. With radicals, it gets a bit more interesting. Using the rule of multiplying under the radical \( \sqrt{6} \times \sqrt{24} \), we find the product as \( \sqrt{144} \). This multiply-and-simplify technique streamlines the handling of roots and exponents together, reducing complex expressions step by step.

Square Root Simplification

When you encounter a square root like \( \sqrt{144} \), it's important to recognize when it can be simplified to a whole number. This happens because 144 is a perfect square, meaning \( 12 \times 12 = 144 \). Therefore, \( \sqrt{144} = 12 \). By recognizing perfect squares, you can simplify expressions quickly and conclude a simplification journey. Thus, our complex original expression eventually simplifies to \( b^{12} \), providing a clean, straightforward result.