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Law of Exponent - 3

by Sharina
(Philippines)











































(-2x5) raise to the poer of 2/(-4x4)raise of the poer of 3=

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Oct 14, 2011
Exponents - Powers of Powers
by: Staff


Question:

by Sharina
(Philippines)


(-2x5) raise to the poer of 2/(-4x4)raise of the poer of 3=


Answer:

(-2x⁵)² / (-4x⁴)³

When you encounter Powers of Powers [ (-2x⁵)² or (-4x⁴)³ ], multiply the exponents


= [(-1)¹ * (2)¹ * (x)⁵]² / [(-1)¹ * (4)¹ * (x)⁴]³

Multiply the exponents.

= [(-1)⁽¹*²⁾ * (2)⁽¹*²⁾ * (x)⁽⁵*²⁾] / [(-1)⁽¹* ³⁾ * (4)⁽¹* ³⁾ * (x)⁽⁴* ³⁾]

= [(-1)⁽²⁾ * (2)⁽²⁾ * (x)⁽¹⁰⁾] / [(-1)⁽³⁾ * (4)⁽³⁾ * (x)⁽¹²⁾]

= [(1) * (4) * (x¹⁰)] / [(-1) * (64) * (x¹²)]

= [(1) * (4) * (x¹⁰)] / [(-1) * (64) * (x¹²)]

= [(-1) * (4) * (x¹⁰)] / [(1) * (64) * (x¹²)]

= (-1) * [(4) * (x¹⁰)] / [(64) * (x¹²)]

= (-1) * [(2²) * (x¹⁰)] / [(2⁶) * (x¹²)]

Subtract the exponents

= (-1) / [(2⁽⁶⁻²⁾) * (x⁽¹²⁻¹⁰⁾)]


= (-1) / [(2⁽⁴⁾) * (x⁽²⁾)]

= (-1) / [(16) * (x²)]

= (-1) / (16x²)

= -1 / 16x²


Another way:

To understand the process in a little more detail, begin by considering the numerator and denominator separately.

Numerator:

(-2x⁵)²

(-2x⁵) is repeated 2 times

= (-2x⁵) * (-2x⁵)

= (-2) * (x⁵) * (-2) * (x⁵)

= (-2) * (-2) * (x⁵) * (x⁵)

= [(-2) * (-2)] * [(x⁵) * (x⁵)]

= [(-2) * (-2)] * [(x * x * x * x * x) * (x * x * x * x * x)]

= (4) * [(x * x * x * x * x) * (x * x * x * x * x)]

= (4) * (x * x * x * x * x * x * x * x * x * x)

x is repeated 10 times

= (4) * (x¹⁰)

= 4 * x¹⁰

= 4x¹⁰


Denominator:

(-4x⁴)³

(-4x⁴) is repeated 3 times


= (-4x⁴)* (-4x⁴)* (-4x⁴)

= (-4) * (x⁴) * (-4) * (x⁴) * (-4) * (x⁴)

= (-4) * (-4) * (-4) * (x⁴) * (x⁴) * (x⁴)

= [(-4) * (-4) * (-4)] * [(x⁴) * (x⁴) * (x⁴)]

= [(-4) * (-4) * (-4)] * [(x * x * x * x) * (x * x * x * x) * (x * x * x * x)]

= (-64) * [(x * x * x * x) * (x * x * x * x) * (x * x * x * x)]

= (-64) * (x * x * x * x * x * x * x * x * x * x * x * x)


x is repeated 12 times

= (-64) * (x¹²)

= -64 * x¹²

= -64x¹²


The original fraction can be rewritten:

(-2x⁵)² / (-4x⁴)³

= 4x¹⁰/ (-64x¹²)


Factor both the numerator and denominator. The common factors can be canceled:

= 4x¹⁰/ (-64x¹²)

= [(2 * 2) * (x * x * x * x * x * x * x * x * x * x)] / [(-1) * (2 * 2 * 2 * 2 * 2 * 2) * (x * x * x * x * x * x * x * x * x * x * x * x)]


Factors which appear in the numerator AND denominator can be canceled:

= {1 / [(-1) * (2 * 2 * 2 * 2) * (x * x)]} * (2 / 2) * (2 / 2) * (x / x) * (x / x) * (x / x) * (x / x) * (x / x) * (x / x) * (x / x) * (x / x) * (x / x) * (x / x)

= {1 / [(-1) * (2 * 2 * 2 * 2) * (x * x)]} * (1) * (1) * (1) * (1) * (1) * (1) * (1) * (1) * (1) * (1) * (1) * (1)

= {1 / [(-1) * (2 * 2 * 2 * 2) * (x * x)]}

= 1 / [(-1) * (2 * 2 * 2 * 2) * (x * x)]

= 1 / [(-16) * (x * x)]

= 1 / [(-16) * (x²)]

= -1 / [(16) * (x²)]

= -1 / (16x²)

= -1 / 16x²


The final answer is: -1 / 16x²


Thanks for writing.

Staff
www.solving-math-problems.com


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