Problem 147 - Elementary and Intermediate Algebra

Question

The fraction \(\frac{2}{4}\) is equal to \(\frac{1}{2} .\) Is \(16^{2 / 4}\) equal to \(16^{1 / 2}\) ? Explain.

Step-by-Step Solution

Answer

Yes, because \( \frac{2}{4} \) simplifies to \( \frac{1}{2} \), so \( 16^{2/4} = 16^{1/2} = 4 \).

Step 1: Simplify the Exponent

First, we simplify the exponent \( \frac{2}{4} \). This fraction can be simplified by dividing both the numerator and the denominator by 2, which gives us \( \frac{1}{2} \). Hence, \( 16^{2/4} = 16^{1/2} \).

Step 2: Evaluate the Expressions

Next, we evaluate both expressions \( 16^{2/4} \) and \( 16^{1/2} \). We have shown in Step 1 that both exponents are the same, so \( 16^{2/4} = 16^{1/2} \).

Step 3: Interpret the Expression

The expression \( 16^{1/2} \) represents the square root of 16. Therefore, \( 16^{1/2} = \sqrt{16} = 4 \).

Step 4: Conclude the Comparison

Since both expressions \( 16^{2/4} \) and \( 16^{1/2} \) simplify to \( 4 \), we conclude that \( 16^{2/4} \) is indeed equal to \( 16^{1/2} \).

Key Concepts

Fractions

Fractions are numerical expressions that represent the division of one quantity by another. They consist of two parts: the numerator, which is the number above the division line, and the denominator, which is the number below the line. For example, in the fraction \( \frac{2}{4} \), 2 is the numerator, and 4 is the denominator. This fraction tells us that we are dividing 2 by 4.

Fractions can often be simplified to make them easier to work with. Simplification involves dividing both the numerator and the denominator by their greatest common divisor (GCD). For instance, the GCD of 2 and 4 is 2. Dividing both by 2 gives \( \frac{1}{2} \), which is the simplified form of \( \frac{2}{4} \).

Using simplified fractions often makes calculations more straightforward and prevents errors, particularly when they appear as exponents, like they do in this exercise.

Simplifying Expressions

When we talk about simplifying expressions, we mean reducing them to their simplest form. This could involve simplifying fractions, like turning \( \frac{2}{4} \) into \( \frac{1}{2} \), or simplifying mathematical expressions through arithmetic operations.

In the context of this exercise, simplifying \( 16^{2/4} \) to \( 16^{1/2} \) is an example of exponent simplification. Once the fraction in the exponent is simplified, the expression itself becomes easier to evaluate.

• Check for the simplest form: Look for common factors between the numerator and denominator.
• Simplify gradually: Ensure each step is straightforward and clear.
• End result should be the simplest possible expression without changing its value.

Simplifying in this way helps achieve a clearer understanding of the problem at hand, and shows how seemingly complex expressions can be made more manageable.

Square Roots

Square roots are a particular kind of root that ask for a number which, when multiplied by itself, gives the original number. In mathematical terms, the square root of 16 is the number that, when squared (multiplied by itself), equals 16.

The square root is often notated by the radical symbol \( \sqrt{ } \). It can also be expressed using exponents as \( x^{1/2} \), where \( x \) is any number. So, \( 16^{1/2} \) is essentially asking for \( \sqrt{16} \), which equals 4.

Calculating square roots can often involve either recognizing perfect squares or using a calculator for non-perfect squares. For example, 16 is a perfect square because \( 4 \times 4 = 16 \). Therefore \( \sqrt{16} = 4 \), supporting that \( 16^{2/4} \) or \( 16^{1/2} \) both correctly equate to 4.