Problem 17 - A Graphical Approach to Precalculus with Limits

Question

Evaluate each expression. $$64^{1 / 6}$$

Step-by-Step Solution

Answer

The value of \( 64^{1/6} \) is 2.

Step 1: Understand the Expression

The expression we need to evaluate is \( 64^{1/6} \). This is a fractional exponent expression, where the base is 64 and the exponent is \( \frac{1}{6} \). The exponent \( \frac{1}{6} \) means we need to find the sixth root of 64.

Step 2: Identify the Sixth Root

To solve \( 64^{1/6} \), we need to determine a number that, when raised to the power of 6, equals 64. Start by testing small integers. First test 2 raised to the power of 6: \( 2^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64 \).

Step 3: Verify the Result

Now verify that \( 2^6 = 64 \) is correct. Calculate again: start with \( 2 \times 2 = 4 \), then \( 4 \times 2 = 8 \), \( 8 \times 2 = 16 \), \( 16 \times 2 = 32 \), and \( 32 \times 2 = 64 \). Since \( 2^6 = 64 \), the sixth root of 64 is indeed 2.

Step 4: Simplify the Answer

Since the sixth root of 64 is 2, we can write \( 64^{1/6} = 2 \).

Key Concepts

Roots of Numbers

In the world of mathematics, understanding roots is essential, especially when dealing with fractional exponents. Roots of numbers are the inverse operation of exponentiation. When you see an expression like \( 64^{1/6} \), it's asking us to find a number that, when raised to the 6th power, gives 64. This is known as the sixth root.

• The most common roots are square roots \((1/2)\), cube roots \((1/3)\), but any positive integer can serve as a root.
• A root's goal is to "undo" the exponentiation of a number.
• Finding roots involves identifying a base number that matches the given condition.

For \( 64^{1/6} \), we identified that the sixth root of 64 is 2, since \( 2^6 = 64 \). This process involves recognizing the relation between exponents and roots.

Exponentiation

Exponentiation is a mathematical operation where a number, known as the base, is multiplied by itself a specific number of times indicated by the exponent. For instance, in the expression \( 2^6 \), 2 is multiplied by itself 6 times.

• The base is the number being multiplied.
• The exponent denotes how many times the base multiplies itself.
• Exponents form the foundation for exponential functions and powers in mathematics.

Understanding fractional exponents, like \( \frac{1}{6} \) in \( 64^{1/6} \), is crucial as they indicate root operations rather than direct multiplication. Fractional exponents help decipher roots and make calculations involving roots efficient and simplified.

Simplification of Expressions

Simplification of expressions involves reducing complex equations into a simpler and more manageable form. With fractions and exponentiation, simplification allows for clearer understanding and problem-solving.

• It helps in finding a straightforward answer or expression.
• Reduction involves breaking down numbers into factors or using arithmetic rules, such as matching powers and roots.
• The goal is to make calculations simpler and less prone to error.

When we simplify \( 64^{1/6} \) to the answer 2, we effectively communicate the original complex expression in a reduced, more understandable form. This process often confirms the correctness of our earlier calculations and aligns with finding roots or powers, ensuring the expression’s accuracy.