What is the simplified value of the expression (256)^{0.16} \times (256)^{0.09}? — State PSC Exam

What is the simplified value of the expression $(256)^{0.16} \times (256)^{0.09}$ ?

The simplified value of

(256)^{0.16} \times (256)^{0.09}

4

According to the basic laws of indices, multiplying terms with equal bases requires adding their exponents:

(256)^{0.16} \times (256)^{0.09} = (256)^{0.16 + 0.09} = (256)^{0.25}

. Expressing

0.25

\frac{1}{4}

converts the expression to the fourth root of

256

, which equals

4

because

4^4 = 256

Apply the law of indices

a^m \times a^n = a^{m+n}

(256)^{0.16 + 0.09} = (256)^{0.25}

When multiplying exponential terms with identical bases, add their powers.

Convert the decimal exponent to a simplified fraction

0.25 = \frac{25}{100} = \frac{1}{4}

Converting to a fraction simplifies root evaluation.

Express

256

as a base with power

4

256 = 4^4

Writing the base as

4^4

allows straightforward simplification with the exponent

\frac{1}{4}

Evaluate

(4^4)^{1/4}

4^{4 \times \frac{1}{4}} = 4^1 = 4

Apply the power rule

(a^m)^n = a^{m \times n}

to arrive at the final simplified value.

Laws of Indices: Product Rule (

a^m \times a^n = a^{m+n}

) and Power of a Power Rule (

(a^m)^n = a^{mn}

)

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