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Which choice is equivalent to the expression below?

[tex]\[ 2^{7.19} \][/tex]

A. [tex]\[ 2^7 \cdot 2^{1 / 10} \cdot 2^{9 / 100} \][/tex]

B. [tex]\[ 2^7 \cdot 2^{19 / 10} \][/tex]

C. [tex]\[ 2^{7+1 / 10+9 / 10} \][/tex]

D. [tex]\[ 2^7+2^{1 / 10}+2^{9 / 100} \][/tex]


Sagot :

To determine which choice is equivalent to the expression [tex]\(2^{7.19}\)[/tex], we need to simplify [tex]\(2^{7.19}\)[/tex] and compare it with each option.

First, let’s rewrite [tex]\(2^{7.19}\)[/tex]:
[tex]\[ 2^{7.19} = 2^{7 + 0.19} \][/tex]

The exponent [tex]\(7.19\)[/tex] can be broken down into [tex]\(7\)[/tex] and [tex]\(0.19\)[/tex]. Then we can use the property of exponents that [tex]\(a^{b+c} = a^b \cdot a^c\)[/tex]:
[tex]\[ 2^{7.19} = 2^7 \cdot 2^{0.19} \][/tex]

Next, we need to analyze the provided choices:

Choice A:
[tex]\[ 2^7 \cdot 2^{1/10} \cdot 2^{9/100} \][/tex]
We can combine the exponents using the product of powers property:
[tex]\[ 2^7 \cdot 2^{1/10} \cdot 2^{9/100} = 2^7 \cdot 2^{10/100} \cdot 2^{9/100} = 2^7 \cdot 2^{(10+9)/100} = 2^7 \cdot 2^{19/100} = 2^{7 + 0.19} \][/tex]

This matches our expression for [tex]\(2^{7.19}\)[/tex], indicating that choice A is an equivalent expression.

Choice B:
[tex]\[ 2^7 \cdot 2^{19/10} \][/tex]
Combine the exponents:
[tex]\[ 2^7 \cdot 2^{19/10} = 2^{7 + 19/10} = 2^{7 + 1.9} \][/tex]
This simplifies to [tex]\(2^{8.9}\)[/tex], which is not the same as [tex]\(2^{7.19}\)[/tex].

Choice C:
[tex]\[ 2^{7 + 1/10 + 9/10} \][/tex]
Simplify the combined exponent:
[tex]\[ 2^{7 + 1/10 + 9/10} = 2^{7 + 1} = 2^8 \][/tex]
This simplifies to [tex]\(2^8\)[/tex], which is not the same as [tex]\(2^{7.19}\)[/tex].

Choice D:
[tex]\[ 2^7 + 2^{1/10} + 2^{9/100} \][/tex]
This is a sum of three terms, not a product. The expression [tex]\(2^{7.19}\)[/tex] cannot be represented as the sum of exponential terms.

Based on the simplification and comparison, the equivalent expression to [tex]\(2^{7.19}\)[/tex] is:

[tex]\[ \boxed{A} \][/tex]