Problem 16 - Algebra for College Students

Question

Evaluate each numerical expression. \(4^{\frac{7}{2}}\)

Step-by-Step Solution

Answer

The value of \(4^{\frac{7}{2}}\) is 128.

Step 1: Understand the expression

The expression given is \(4^{\frac{7}{2}}\). This is a fractional exponent, where the numerator is the power and the denominator is the root.

Step 2: Simplify using the root

A fractional exponent \(\frac{m}{n}\) can be expressed as the nth root. So, \(4^{\frac{7}{2}}\) is the square root of \(4^7\). Start by finding \(\sqrt{4}\) which is 2.

Step 3: Compute \(4^7\)

Calculate \(4^7\). Multiply the base, 4, by itself 6 more times: \(4 \times 4 = 16\), \(16 \times 4 = 64\), \(64 \times 4 = 256\), \(256 \times 4 = 1024\), \(1024 \times 4 = 4096\), and \(4096 \times 4 = 16384\).

Step 4: Apply the square root

Now find the square root of 16384. Since we previously found \(\sqrt{4} = 2\), the answer can be derived by rearranging the expression \(\sqrt{16384} = 4^3\). Therefore, \(4^{\frac{7}{2}} = (4^7)^{\frac{1}{2}} = \sqrt{16384} = 128\).

Step 5: Verify the final answer

Ensure the calculations are correct: \(4^7 = 16384\), and \(\sqrt{16384} = 128\). Hence, \(4^{\frac{7}{2}} = 128\). This confirms our solution is accurate.

Key Concepts

Understanding the Square Root

The square root is a fundamental concept in mathematics. It is the value that, when multiplied by itself, gives the original number. For example, the square root of 4 is 2 because

• 2 multiplied by 2 equals 4.

This relationship is typically represented by the symbol \( \sqrt{} \), so \( \sqrt{4} = 2 \). Working with square roots is essential when dealing with fractional exponents, as these often involve taking the square root of a number raised to a power. It's crucial to understand that finding a square root is essentially asking, "What number times itself equals this number?" Square roots are used extensively in both theoretical and applied mathematics.

What is Exponentiation?

Exponentiation refers to the mathematical operation of raising a number, known as the base, to an exponent. This is a repeated multiplication. For instance, in \(4^7\), 4 is the base and 7 is the exponent. This means we multiply 4 by itself 7 times:

• \(4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4\).

The act of exponentiation is crucial to working with powers of numbers and is often used in calculating large quantities or simplifying expressions. When paired with fractional exponents, exponentiation helps us understand how properties like roots are derived from powers, blending multiplication and division into one coherent operation.

Simplification in Mathematics

Simplification is the process of reducing an expression to its simplest form, making it easier to work with. This involves combining like terms and removing any unnecessary parts to reveal a clearer solution.

• Simplifying \(4^{\frac{7}{2}}\) requires understanding that this can be broken into two tasks—raising 4 to the 7th power and then taking the square root.
• This makes \(4^{\frac{7}{2}} = \sqrt{4^7}\).

After calculating \(4^7 = 16384\), simplifying further involves calculating its square root. This approach not only makes the problem manageable but also underscores the elegance of mathematical principles, where complex problems become simple through well-defined steps.

Breaking Down Numerical Expressions

A numerical expression is a mathematical phrase involving numbers and operations like addition, subtraction, multiplication, and division. In this exercise, \(4^{\frac{7}{2}}\) is a numerical expression that incorporates an exponent and a root.

• To evaluate such expressions, follow order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

Understanding the nature of the expression helps in systematically tackling each part. For \(4^{\frac{7}{2}}\), recognizing the use of a fractional exponent is key. This tells us that we are handling large numbers and roots, and by breaking down the expression into parts—powers first, followed by roots—we are ensuring clarity and correctness in our final solution.