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# M06-20

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Math Expert
Joined: 02 Sep 2009
Posts: 58435

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16 Sep 2014, 00:27
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25% (medium)

Question Stats:

61% (00:35) correct 39% (00:41) wrong based on 242 sessions

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The square of $$5^{\sqrt{2}}$$ = ?

A. $$5^2$$
B. $$25^{\sqrt{2}}$$
C. $$25$$
D. $$25^{2\sqrt{2}}$$
E. $$5^{\sqrt{2}^2}$$

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Math Expert
Joined: 02 Sep 2009
Posts: 58435

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16 Sep 2014, 00:28
Official Solution:

The square of $$5^{\sqrt{2}}$$ = ?

A. $$5^2$$
B. $$25^{\sqrt{2}}$$
C. $$25$$
D. $$25^{2\sqrt{2}}$$
E. $$5^{\sqrt{2}^2}$$

$$(5^{\sqrt{2}})^2=5^{2*\sqrt{2}}=(5^2)^{\sqrt{2}}=25^{\sqrt{2}}$$.

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Joined: 30 Jun 2012
Posts: 12

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26 Nov 2014, 14:56
My assumption is the following

5^sqt 2 * 5 sqt = 25 ^ 2 why is it 25 ^ sqt 2?
Math Expert
Joined: 02 Sep 2009
Posts: 58435

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27 Nov 2014, 03:23
1
rsamant wrote:
My assumption is the following

5^sqt 2 * 5 sqt = 25 ^ 2 why is it 25 ^ sqt 2?

First of all read Writing Mathematical Formulas on the Forum.

Also, can you please tell me which step in the following solution is unclear: $$(5^{\sqrt{2}})^2=5^{2*\sqrt{2}}=(5^2)^{\sqrt{2}}=25^{\sqrt{2}}$$. Thank you.
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Joined: 17 Jul 2018
Posts: 16
GMAT 1: 710 Q47 V39
GPA: 3.51

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23 Oct 2018, 19:53
Bunuel wrote:
Official Solution:

The square of $$5^{\sqrt{2}}$$ = ?

A. $$5^2$$
B. $$25^{\sqrt{2}}$$
C. $$25$$
D. $$25^{2\sqrt{2}}$$
E. $$5^{\sqrt{2}^2}$$

$$(5^{\sqrt{2}})^2=5^{2*\sqrt{2}}=(5^2)^{\sqrt{2}}=25^{\sqrt{2}}$$.

If you have the same bases, you add the exponents...why is this different?
Math Expert
Joined: 02 Sep 2009
Posts: 58435

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23 Oct 2018, 20:42
1
1
leonsandcastle332 wrote:
Bunuel wrote:
Official Solution:

The square of $$5^{\sqrt{2}}$$ = ?

A. $$5^2$$
B. $$25^{\sqrt{2}}$$
C. $$25$$
D. $$25^{2\sqrt{2}}$$
E. $$5^{\sqrt{2}^2}$$

$$(5^{\sqrt{2}})^2=5^{2*\sqrt{2}}=(5^2)^{\sqrt{2}}=25^{\sqrt{2}}$$.

If you have the same bases, you add the exponents...why is this different?

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:
$$a^{\frac{1}{n}}=\sqrt[n]{a}$$

$$a^{\frac{m}{n}}=\sqrt[n]{a^m}$$

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 $$\frac{\sqrt{x}}{\sqrt{y}}=\sqrt{\frac{x}{y}}$$.

• $$(\sqrt{x})^n=\sqrt{x^n}$$

• $$x^{\frac{1}{n}}=\sqrt[n]{x}$$

• $$x^{\frac{n}{m}}=\sqrt[m]{x^n}$$

• $${\sqrt{a}}+{\sqrt{b}}\neq{\sqrt{a+b}}$$

• $$\sqrt{x^2}=|x|$$, 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 $$\sqrt[4]{x}$$, 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, $$\sqrt[3]{125} =5$$ and $$\sqrt[3]{-64} =-4$$.

8. Exponents and Roots of Numbers

Check below for more:
ALL YOU NEED FOR QUANT ! ! !

Hope it helps.
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Re: M06-20   [#permalink] 23 Oct 2018, 20:42
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# M06-20

Moderators: chetan2u, Bunuel