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Difficulty: EasyExponents, Powers, and Square Roots

What is the value of 45+45+45+45\sqrt{4^5 + 4^5 + 4^5 + 4^5}?

  1. A
    16
  2. 64Answer
  3. C
    128
  4. D
    256
  5. E
    1024

Answer

64
Combining the four terms inside the radical gives 4×45=464 \times 4^5 = 4^6. Taking the square root yields 46=43\sqrt{4^6} = 4^3, which evaluates to 64.

Step-by-Step Solution

1
Rewrite the sum inside the radical as multiplication
45+45+45+45=4×454^5 + 4^5 + 4^5 + 4^5 = 4 \times 4^5
Adding four identical terms is equivalent to multiplying the term by 4.
2
Apply the product rule for exponents
41×45=41+5=464^1 \times 4^5 = 4^{1+5} = 4^6
When multiplying exponential expressions with the same base, add their exponents.
3
Simplify the radical expression
46=(46)1/2=43=64\sqrt{4^6} = (4^6)^{1/2} = 4^3 = 64
Taking the square root of a power is equivalent to dividing the exponent by 2.

Key Concept

Exponent Addition and Radical Laws
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Topics

Exponents, Powers, and Square Roots

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