Powers w/Rational Exponents: Evaluate (Review My Work)

So the problem reduces to 2^11=2048.In summary, the first problem simplifies to -1/8 while the second problem simplifies to 2048.
  • #1
calcdummy
11
0

Homework Statement


Write as a single power, then evaluate:
a) (-32)^3/5 x (-32)^-4/5 / (-32)^2/5


b) 4096^3/6 / 4096^2/3 x 4096^5/6

Homework Equations





The Attempt at a Solution



a) (-32)^3/5 x (-32)^-4/5 / (-32)^2/5
= (-32)^3/5+(-4/5)-2/5
= -32^-3/5
= -1/8 <- not sure about this

b) 4096^3/6 / 4096^2/3 x 4096^5/6
= 4096^(9-8+10)/12
= 4096^11/12

I'm not so sure of where to go from here.
 
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  • #2


Your first answer is right :-)

Second needs some review.

Remember (x^a*x^b)/(x^c*x^d)

Is x^(a+b-c-d)

So its 4096^(3/6-2/3-5/6)


You have made a mistake by writing 4096^({9}-8-10)/12

It shouldn't be 9 in the curly brackets i put{} :-).

Also factorise 4096.
Write it in power of primes.

For eg 400=2^4*5^2

So (400)^(1/2)

Is
[(2^4)*(5^2)]^(1/2)

So its (2^2)*(5) which gives 20.
 
Last edited:
  • #3


Oh man I copied down the wrong problem. I'm so sorry. It was supposed to be:
4096^3/4 / 4096^2/3 x 4096^5/6
I got the common denominator which would been 12. That is how I got 4096^(9-8+10)/12
= 4096^11/12

am I still incorrect?
 
  • #4


calcdummy said:
Oh man I copied down the wrong problem. I'm so sorry. It was supposed to be:
4096^3/4 / 4096^2/3 x 4096^5/6
I got the common denominator which would been 12. That is how I got 4096^(9-8+10)/12
= 4096^11/12

am I still incorrect?

You should probably use parentheses instead of spaces to make that clearer. If you mean (4096^(3/4)/4096^(2/3))*4096^(5/6) then 4096^(11/12) is correct. There's a much simpler way to express that answer. 4096^(1/12)=2.
 

FAQ: Powers w/Rational Exponents: Evaluate (Review My Work)

What is a rational exponent?

A rational exponent is an exponent in the form of a fraction. It represents a power or root of a number. For example, the exponent 1/2 represents the square root of a number, while the exponent 1/3 represents the cube root of a number.

How do you evaluate powers with rational exponents?

To evaluate powers with rational exponents, you can either convert the exponent to a radical form and simplify, or use the power rule for exponents. For example, if you have the expression 16^(3/4), you can rewrite it as the fourth root of 16 cubed, which is equal to 8. Alternatively, you can use the power rule and calculate 16^(3/4) as (16^3)^(1/4), which is also equal to 8.

What is the difference between rational exponents and radical expressions?

Rational exponents and radical expressions are different ways of representing the same mathematical concept. Rational exponents use fractional exponents, while radical expressions use roots. For example, 8^(1/3) is equivalent to the cube root of 8, or ∛8.

Can rational exponents be negative?

Yes, rational exponents can be negative. A negative exponent indicates the reciprocal of the base raised to the positive exponent. For example, 2^(-1/2) is equal to 1/(2^(1/2)), which is the same as 1/√2.

How do you simplify expressions with rational exponents?

To simplify expressions with rational exponents, you can use the laws of exponents, such as the power rule, product rule, and quotient rule. You can also use the rules of radicals, such as the product and quotient rules for radicals. It is important to simplify the expression as much as possible, using any applicable rules, to get the most simplified form.

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