# Simplify using the exponent laws #2

• Sep 6th 2012, 10:52 AM
ford2008
Simplify using the exponent laws #2
Hi guys,

the task: Simplify using the exponent laws. Write the final answer using only positive exponents. Evaluate where possible.

1. 2x^-1

2
---
x

Is that right?

2. (5p^3)^-2

1
-----
(5p^3)^2

= 1
-------
10p^6

Is that right?

3. (-32x^-10 y^15)^1/5

(-32)^1/5 (x^-10)^1/5 (y^15)^1/5

How can I simplify further???

4. ( x^1/2 ) ^1/2
(------------)
(x^1/3 x^1/5)

the whole task is in one bracket and ^1/2 is outside the bracket.
How can I simplify?
• Sep 6th 2012, 11:14 AM
earboth
Re: Simplify using the exponent laws #2
Quote:

Originally Posted by ford2008
Hi guys,

the task: Simplify using the exponent laws. Write the final answer using only positive exponents. Evaluate where possible.

1. 2x^-1

2
---
x

Is that right? <--- yes

2. (5p^3)^-2

1
-----
(5p^3)^2

= 1
-------
10p^6

Is that right? <--- no, because 5^2 = 25

3. (-32x^-10 y^15)^1/5

(-32)^1/5 (x^-10)^1/5 (y^15)^1/5

How can I simplify further??? <--- The exponents must be multiplied. ... and 32 = 2^5

4. ( x^1/2 ) ^1/2
(------------)
(x^1/3 x^1/5)

the whole task is in one bracket and ^1/2 is outside the bracket.
How can I simplify? <--- Simplify first the term in the large bracket:

$\left(\frac{x^\frac12}{x^\frac13 \cdot x^\frac15}\right)^\frac12=\left(\frac{x^\frac12}{x ^\frac8{15}}\right)^\frac12 = \left({x^{-\frac1{30}}\right)^\frac12$

See my remark at #3.
• Sep 6th 2012, 11:32 AM
ford2008
Re: Simplify using the exponent laws #2
could you please explain more no. 3?
• Sep 6th 2012, 12:09 PM
HallsofIvy
Re: Simplify using the exponent laws #2
Quote:

Originally Posted by ford2008
Hi guys,

the task: Simplify using the exponent laws. Write the final answer using only positive exponents. Evaluate where possible.

1. 2x^-1

2
---
x

Is that right?

2. (5p^3)^-2

1
-----
(5p^3)^2

= 1
-------
10p^6

No, it is not. 5^2= 25, not 10.

Quote:

Is that right?

3. (-32x^-10 y^15)^1/5

(-32)^1/5 (x^-10)^1/5 (y^15)^1/5

How can I simplify further???
Why not use the laws of exponents as you did before? (x^-10)^1/5= x^(-10/5) and (y^15)^(1/5)= y^(15/5). And you should be able to calculate that (-2)^2= 4, (-2)^3= (4)(-2)= -8, (-2)^4= (-8)(-2)= 16, and (-2)^5= (16)(-2)= -32.

Quote:

4. ( x^1/2 ) ^1/2
(------------)
(x^1/3 x^1/5)

the whole task is in one bracket and ^1/2 is outside the bracket.
How can I simplify?
Again, use the "laws of exponents": x^a x^b= x^(a+b), (x^a/x^b)= x^(a- b), and (x^a)^b= x^(ab). For example the denominator is x^(1/3)x^(1/5)= x^(1/3+ 1/5). What is 1/3+ 1/5?
• Sep 6th 2012, 12:30 PM
ford2008
Re: Simplify using the exponent laws #2
So the answer for 3 should be = -2^5 x^-10/5 y^3 or not? I am just confused because I have to write the final answer using only positive exponents. But the x exponent is still negative or not?

Task 4 = 1/3 + 1/5 is 5/15 + 3/15 = 8/15.
• Sep 6th 2012, 10:38 PM
earboth
Re: Simplify using the exponent laws #2
Quote:

Originally Posted by ford2008
So the answer for 3 should be = -2^5 x^-10/5 y^3 <--- nearly! or not? I am just confused because I have to write the final answer using only positive exponents. But the x exponent is still negative or not?

Task 4 = 1/3 + 1/5 is 5/15 + 3/15 = 8/15.

At #3:

$\left(-32 \cdot x^{-10} \cdot y^{15} \right)^\frac15 = \left(-2^5 \cdot x^{-10} \cdot y^{15} \right)^\frac15 = -2^{5\cdot \frac15} \cdot x^{-10\cdot \frac15} \cdot y^{15\cdot \frac15}$

Now simplify. (btw: -10/5 = -2)
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