Problem 48 - Algebra 2

Question

Simplify each expression. $$ \sqrt[3]{\sqrt{8}} $$

Step-by-Step Solution

Answer

\( \sqrt[3]{\sqrt{8}} = \sqrt{2} \).

Step 1: Evaluate the Innermost Expression

Within the expression \( \sqrt[3]{\sqrt{8}} \), we first consider \( \sqrt{8} \). The number 8 can be expressed as \( 2^3 \). Therefore, \( \sqrt{8} = \sqrt{2^3} \). To evaluate \( \sqrt{2^3} \), remember that taking a square root is equivalent to raising to the power of \( \frac{1}{2} \). Thus, \( \sqrt{8} = (2^3)^{\frac{1}{2}} = 2^{\frac{3}{2}} \).

Step 2: Substitute and Simplify the Outer Expression

Now substitute \( \sqrt{8} = 2^{\frac{3}{2}} \) back into the original expression \( \sqrt[3]{\sqrt{8}} \) resulting in \( \sqrt[3]{2^{\frac{3}{2}}} \). Remember, taking the cube root of a number is equivalent to raising that number to the power of \( \frac{1}{3} \). This gives us \( (2^{\frac{3}{2}})^{\frac{1}{3}} \).

Step 3: Apply the Power of a Power Property

Using the power of a power property \((a^m)^n = a^{m \cdot n}\), we simplify \((2^{\frac{3}{2}})^{\frac{1}{3}} = 2^{\frac{3}{2} \cdot \frac{1}{3}}\). Multiply the exponents: \(\frac{3}{2} \cdot \frac{1}{3} = \frac{3}{6} = \frac{1}{2}\).

Step 4: Simplify the Exponent

Thus, the expression simplifies to \(2^{\frac{1}{2}}\). Recall that raising a number to the power of \(\frac{1}{2}\) is the definition of taking the square root. Therefore, \(2^{\frac{1}{2}} = \sqrt{2}\).

Key Concepts

Exponents

Exponents are a fundamental concept in algebra and mathematics. They indicate how many times a number, known as the base, is multiplied by itself. For example, in the term \(2^3\), 2 is the base and 3 is the exponent, meaning the base 2 is multiplied by itself three times (\(2 \times 2 \times 2\)).
Exponents can be whole numbers, fractions, or even negative numbers, and each of these different kinds of exponents tells us to perform a different mathematical operation:

• Whole number exponents tell us to multiply the base by itself as many times as the exponent indicates.
• Fractional exponents are another way to write roots; for instance, \(x^{\frac{1}{n}}\) means the \(n\)-th root of \(x\).
• Negative exponents indicate the reciprocal of the base with the opposite positive exponent, like \(x^{-n} = \frac{1}{x^n}\).

Understanding how to manipulate exponents, including multiplying and dividing them, is a key skill in simplifying algebraic expressions.

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. A radical is written using the radical sign (\(\sqrt{}\)). For example, the expression \( \sqrt{8} \) is a radical expression that represents the principal square root of 8.
Simplifying radical expressions often involves transforming the expression by using properties of exponents. For instance, a square root can be expressed as a power of \(\frac{1}{2}\), as in \(\sqrt{x} = x^{\frac{1}{2}}\).
When dealing with nested radicals, or radicals within radicals such as \(\sqrt[3]{\sqrt{8}}\), simplifying involves working from the inside out. First, simplify the innermost radical expression, then proceed to the outer layers. Knowing how to convert radicals to exponential form can make this process more straightforward, especially when the expression needs further simplification.
Key steps often include factoring the number under the radical to find perfect squares (or cubes for cube roots), rewriting the expression using fractional exponents, and then simplifying.

Power of a Power Property

The power of a power property is an important rule when dealing with exponents. It states that when raising an exponent to another exponent, you multiply the exponents. Mathematically, for any base \(a\), exponent \(m\), and exponent \(n\), the property is expressed as \((a^m)^n = a^{m \cdot n}\).
This property is particularly useful in simplifying expressions with multiple layers of exponents, like in nested radicals. For example, consider the expression \((2^{\frac{3}{2}})^{\frac{1}{3}}\). Using the power of a power property, we multiply the exponents: \(2^{\frac{3}{2} \cdot \frac{1}{3}}\). This simplifies to \(2^{\frac{1}{2}}\).
This simplification step results in a much simpler expression which can then be interpreted as a root. In this case, \(2^{\frac{1}{2}}\) signifies the square root of 2.
Understanding this property aids in breaking down complex expressions, allowing each part of the expression to be simplified systematically.