Why is 16^(1/2) Equal to 4?

Why is 16^(1/2) Equal to 4?

The expression 16^(1/2) means finding the square root of 16. The square root of a number is a value that, when multiplied by itself, equals the original number. In this case, 4 multiplied by itself (4 x 4) equals 16, so 16^(1/2) is indeed equal to 4.

Understanding Fractional Exponents with Square Roots

In algebra, exponents indicate repeated multiplication of a base number. When the exponent is a fraction, it represents a root operation rather than repeated multiplication. This is why an expression like 16^(1/2) equals 4 rather than 16 – the 1/2 exponent indicates a square root operation.

Let’s explore why 16^(1/2) = 4 in detail:

Exponent Rules

Some key rules of exponents:

• 16^2 = 16 x 16 = 256
• 16^1 = 16
• 16^0 = 1

These examples follow the pattern of multiplying 16 by itself the number of times indicated by the exponent.

Fractional Exponents

However, fractional exponents like 1/2 don’t represent repeated multiplication. Instead:

• 16^(1/2) is read as “the square root of 16”
• The 1/2 exponent means to take the second root, or square root

This relates exponents to roots as inverse operations.

Simplifying the Square Root

If we calculate √16 using the definition of a square root:

√16 = some number that, when squared, equals 16

That number is 4, because:

4^2 = 4 x 4 = 16

So √16 = 4. This matches the result from the fractional exponent form 16^(1/2).

In summary, fractional exponents provide compact notation for roots. The denominator indicates the root degree, while the numerator keeps the expression equivalent. Understanding this connects exponents and roots, explaining why 16^(1/2) = 4 based on exponent and root rules.