MULTIPLY EXPONENTS -
POWER OF A POWER
EQUAL TO BASE POWER
PRODUCT OF EXPONENTS
EXAMPLES, EXERCISE

Please study
RATIONAL EXPONENTS BEFORE MULTIPLY EXPONENTS
if you have not already done so.

It is a prerequisite here.

There, we provided the explanation
for Rational Exponents.

We discussed how we can apply the same
7 Laws and the 2 Rules given for
whole number Exponents can be applied
for Fractional Exponents.

Here, we apply the Laws to solve
problems with emphasis on the law
exponent of a power to some base
= multiply exponents to that base.

Solved Example 1 of Multiply Exponents

Simplify {(27)^2n⁄3 x (8)^-n⁄6}⁄{(18)^-n⁄2}

Solution to Example 1 of Multiply Exponents:

Let A = {(27)^2n⁄3 x (8)^-n⁄6}⁄{(18)^-n⁄2}
We know
27 = 3 x 3 x3 = 3³; 8 = 2 x 2 x 2 = 2³; 18 = 2 x 9 = 2 x 3 x 3 = 2 x 3²
∴ A = {(3³)^2n⁄3 x (2³)^-n⁄6}⁄{(2 x 3²)^-n⁄2}
We Know
(a^m)ⁿ = a^mn]
(ab)^m = a^m x b^m
Applying these, we get
A = {(3^{3 x 2n⁄3}) x (2^{3 x -n⁄6})}⁄{(2^-n⁄2) x (3²)^-n⁄2}
= {(3²ⁿ) x (2^-n⁄2)}⁄{(2^-n⁄2) x (3^{2 x -n⁄2})}
= (3²ⁿ)⁄(3^-n) [By cancelling (2^-n⁄2) which is in Numerator and denominator]
= (3^{2n - (-n)}) = (3^{2n + n}) = 3³ⁿ. Ans.

Solved Example 2 of Multiply Exponents

Prove that (x^a⁄x^b)^{(a + b)}.(x^b⁄x^c)^{(b + c)}.(x^c⁄x^a)^{(c + a)} = 1.

Solution to Example 2 of Multiply Exponents:

We know a^m⁄aⁿ = a^{m - n}
And this can be applied to fractional Exponents also.
Applying this here, we get
L.H.S. = (x^{a - b})^{(a + b)}.(x^{b - c})^{(b + c)}.(x^{c - a})^{(c + a)}.
We Know
(a^m)ⁿ = a^mn
And this can be applied to fractional Exponents also.
Applying this here, we get
L.H.S. = {x^{(a - b)(a + b)}}.{x^{(b - c)(b + c)}}.{x^{(c - a)(c + a)}}
We know
(a - b)(a + b) = a² - b²; (b - c)(b + c) = b² - c²; (c - a)(c + a) = c² - a²;
∴ L.H.S. = {x^{(a2 - b2)}}.{x^{(b2 - c2)}}.{x^{(c2 - a2)}}
We Know
a^m x aⁿ = a^{m + n}
And this can be applied to fractional Exponents also.
Applying this here, we get
L.H.S. = x^{{(a2 - b2) + (b2 - c2) + (c2 - a2)}}.
= x^{{(a2 - a2) + (b2 - b2) + (c2 - c2)}}.
= x^{{0 + 0 + 0}}. = x⁰ = 1 = R.H.S. (proved)

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Solved Example 3 of Multiply Exponents

If a^x = b, b^y = c, c^z = a, show that xyz = 1.

Solution to Example 3 of Multiply Exponents:

a^x = b .........(i); b^y = c .........(ii); c^z = a.........(iii)
Using (iii) in (i), we get
a^x = b ⇒ (c^z)^x = b ⇒ c^zx = b [Since (a^m)ⁿ = a^mn]
Using (ii) here, we get
c^zx = b ⇒ (b^y)^zx = b ⇒ b^yzx = b
⇒ b^xyz = b¹ ⇒ xyz = 1. (Proved.)

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Solved Example 4 of Multiply Exponents

If (64)^x = 1⁄(256)^y, show that 3x + 4y = 0.

Solution to Example 4 of Multiply Exponents:

By data (64)^x = 1⁄(256)^y
We can see that 64 and 256 are powers of 4.
64 = 4 x 4 x 4 = 4³; 256 = 4 x 4 x 4 x 4 = 4⁴;
(64)^x = 1⁄(256)^y ⇒ (64)^x = (256)^-y ⇒ (4³)^x = (4⁴)^-y
We Know
(a^m)ⁿ = a^mn
Applying this here, we get
4^3x = 4^{4 x -y} ⇒ 4^3x = 4^-4y
Since the bases are same, the exponents have to be equal.
∴ 3x = -4y ⇒ 3x + 4y = 0. (Proved.)

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Exercise on Multiply Exponents

1. Simplify: {7^{(2n + 3)} - (49)^{(n + 2)}}⁄{(343)^{(n + 1)}}^2⁄3
2. (243)^-3⁄5

For Answers, see at the bottom of the page.

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Answers to Exercise on Multiply Exponents

1. -42.