Which of the following is closest to the value of (2^23)(5^26)?

Question

A. 10^23
B. 10^24
C. 10^25
D. 10^26
E. 10^27

broall · Accepted Answer

2^{23} 	imes 5^{26}=2^{23} 	imes 5^{23} 	imes 5^3=10^{23} 	imes 125 \approx 10^{23} 	imes 100 = 10^{25}

The answer is C.

Bunuel · Answer

EXPONENTS

Exponents are a "shortcut" method of showing a number that was multiplied by itself several times. For instance, number  a  multiplied  n  times can be written as  a^n , where  a  represents the base, the number that is multiplied by itself  n  times and  n  represents the exponent. The exponent indicates how many times to multiple the base,  a , by itself.

Exponents one and zero: 
 a^0=1  Any nonzero number to the power of 0 is 1.
For example:  5^0=1  and  (-3)^0=1 
 • Note: the case of 0^0 is not tested on the GMAT.

a^1=a  Any number to the power 1 is itself.

Powers of zero: 
If the exponent is positive, the power of zero is zero:  0^n = 0 , where  n > 0 .

If the exponent is negative, the power of zero ( 0^n , where  n < 0 ) is undefined, because division by zero is implied.

Powers of one: 
 1^n=1  The integer powers of one are one.

Negative powers: 
 a^{-n}=\frac{1}{a^n}

Powers of minus one: 
If n is an even integer, then  (-1)^n=1 .

If n is an odd integer, then  (-1)^n =-1 .

Operations involving the same exponents: 
 Keep the exponent, multiply or divide the bases
 a^n*b^n=(ab)^n

\frac{a^n}{b^n}=(\frac{a}{b})^n

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

a^m^n=a^{(m^n)}  and not  (a^m)^n  (if exponentiation is indicated by stacked symbols, the rule is to work from the top down)

Operations involving the same bases:

Keep the base, add or subtract the exponent (add for multiplication, subtract for division)
 a^n*a^m=a^{n+m}

\frac{a^n}{a^m}=a^{n-m}

Fraction as power: 
 a^{\frac{1}{n}}=\sqrt {a}

a^{\frac{m}{n}}=\sqrt {a^m}

ROOTS

Roots (or radicals) are the "opposite" operation of applying exponents. For instance x^2=16 and square root of 16=4.

General rules: 
•  \sqrt{x}\sqrt{y}=\sqrt{xy}  and  \frac{\sqrt{x}}{\sqrt{y}}=\sqrt{\frac{x}{y}} .

•  (\sqrt{x})^n=\sqrt{x^n}

•  x^{\frac{1}{n}}=\sqrt {x}

•  x^{\frac{n}{m}}=\sqrt {x^n}

•  {\sqrt{a}}+{\sqrt{b}}
eq{\sqrt{a+b}}

•  \sqrt{x^2}=|x| , when  x\leq{0} , then  \sqrt{x^2}=-x  and when  x\geq{0} , then  \sqrt{x^2}=x

• When the GMAT provides the square root sign for an even root, such as  \sqrt{x}  or  \sqrt {x} , then the only accepted answer is the positive root.

That is,  \sqrt{25}=5 , NOT +5 or -5. In contrast, the equation  x^2=25  has TWO solutions, +5 and -5.  Even roots have only a positive value on the GMAT.

• Odd roots will have the same sign as the base of the root. For example,  \sqrt {125} =5  and  \sqrt {-64} =-4 .

8. Exponents and Roots of Numbers 
     Theory   
 Exponents and Roots on the GMAT: Tips and hints 
 Cyclicity and the remainders on the GMAT 
     Questions   
 Units digits, exponents, remainders problems 
 Tough and tricky DS exponents and roots questions with detailed solutions 
 Tough and tricky PS exponents and roots questions with detailed solutions 
 Exponents DS Questions 
 Exponents PS Questions 
 Roots DS Questions 
 Roots PS Questions

Check below for more:
 ALL YOU NEED FOR QUANT ! ! ! 
 Ultimate GMAT Quantitative Megathread

loneo3006 · Answer

Hi guys,

Just wanted to check the rule here with you.
"When two numbers have the same exponent, I can multiply the numbers together and keep the exponent as it was for the two numbers." Is this correct?

loneo3006 · Answer

Thanks,

this was the one I was looking for :

Operations involving the same exponents:
 Keep the exponent, multiply or divide the bases

JeffTargetTestPrep · Answer

Simplifying, we have:

2^23 x 5^23 x 5^3

10^23 x 5^3, which is about 10^23 x 10^2 = 10^25.

Answer: C

GMATWhizTeam · Answer

x^m  *  y^m  =  (xy)^m ; this is the law of exponents that can make questions like these, very easy to solve. This law comes in handy when it comes to dealing with questions on roots too.

2^{23}  *  5^{26}  =  2^{23}  *  5^{23}  *  5^3

Applying the law of exponents,  2^{23}  *  5^{23}  =  (2*5)^{23}  =  10^{23} .

But, hold on! Answer option A is a trap. Although the word closest is mentioned here, it does not mean that you ignore the  5^3 ; that is because  5^3  = 125 and hence contributes a  10^2  to the product.

Therefore, the expression is closest to  10^{25}

The correct answer option is C.

Which of the following is closest to the value of (2^23)(5^26)?

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GMAT Problem Solving (PS) Questions

Which of the following is closest to the value of (2^23)(5^26)?

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