The value of (256)^0.16(256)^0.09 is

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Detailed Solution

The question is asking to calculate the value of (256)^0.16 × (256)^0.09, and the solution involves simplifying the expression using the laws of exponents.

Detailed Explanation

Given Expression:

(256)^0.16 × (256)^0.09

Step 1: Apply the Law of Exponents

The law of exponents states that for a common base:

a^m × aⁿ = a^m+n

Here, the base 256 is common, so the exponents 0.16 and 0.09 can be added:

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

Step 2: Add the Exponents

0.16 + 0.09 = 0.25

So, the expression becomes:

(256)^0.25

Step 3: Simplify the Exponent

The exponent 0.25 can be rewritten as 1/4:

(256)^0.25 = (256)^1/4

Step 4: Find the Fourth Root of 256

The fourth root of 256 is:

√⁴(256) = 4

Final Answer:

The value of (256)^0.16 × (256)^0.09 is:4

Explanation

The image shows the detailed steps of solving (256)^0.16 × (256)^0.09:

Combine Exponents:The exponents 0.16 and 0.09 are added using the law of exponents:(256)^0.16 × (256)^0.09 = (256)^0.25
Rewrite the Exponent:The exponent 0.25 is expressed as 1/4, leading to:(256)^0.25 = (256)^1/4
Simplify the Root:The fourth root of 256 is calculated as:√⁴(256) = 4