On simplifying [tex]8^3 \times 2^4[/tex], which of the following is correct?
(a) [tex]16^7[/tex]
(b) [tex]2^{13}[/tex]
(c) [tex]2^{10}[/tex]
(d) [tex]8^4[/tex]
Answer (2)
The expression 8 3 × 2 4 simplifies to 2 13 by converting 8 to base 2 and combining exponents. Therefore, the correct answer is (b) 2 13 .
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To determine the value of 8^3 \times 2^4 , we need to simplify this expression using basic exponent rules.
Step 1: Simplify each base expression.
8 is equal to 2^3. Therefore, we can rewrite 8 as a power of 2: 8 = 2 3 .
Rewrite 8^3 as (2^3)^3. So, 8 3 = ( 2 3 ) 3 .
Use the power of a power property. This property states that ( a m ) n = a m × n . Applying this to our expression: ( 2 3 ) 3 = 2 3 × 3 = 2 9 .
Step 2: Simplify 2^4 separately.
Leave 2^4 as it is. 2 4 = 2 4 .
Step 3: Multiply the powers of 2.
Using the multiplication rule for exponents with the same base: The rule states that a m × a n = a m + n . So, 2 9 × 2 4 = 2 9 + 4 = 2 13 .
Therefore, after simplifying 8^3 times 2^4, we find that the correct expression is 2 13 .
The correct option is (b) 2^{13}.
