Solving Negative Exponents

In summary, when solving equations involving negative and fractional exponents, you can raise both sides to a power to simplify the equation. This can be helpful in understanding and solving more challenging problems.
  • #1
TbbZz
37
0

Homework Statement


8x^-3 = 64

Homework Equations


None.

The Attempt at a Solution


I tried doing all sorts of things, changing 8x^-3 to (1/8x)^3 or trying to get both sides to have the same base, but couldn't get it to work.

The book I am using does not explain how to do so, I have already looked through the whole chapter that the problem is from.

Primarily, I would like assistance in understanding the concepts behind this problem. With the current problem, it is relatively easy to figure out by educated guessing & checking, but when there are more difficult numbers and more challenging problems, guessing and checking won't work. Thanks for the assistance.
 
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  • #2
Hint: (x^-3)^3 = ?
 
  • #3
berkeman said:
Hint: (x^-3)^3 = ?

(x^-3)^3 = X^-9 but I'm not quite sure how that helps.

Thanks for the response.
 
  • #4
Berkeman meant (x^-3)^(-1/3).
 
  • #5
Got it. I didn't realize you could raise both sides to a power ((-1/3) in this case). Thanks for the help.
 
  • #6
Dick said:
Berkeman meant (x^-3)^(-1/3).

Whoops. Thanks Dick.
 
  • #7
Have you tried dividing both sides by 8?

Then did you try expressing x^-3 as an expression with a positive exponent using the rule a^-b = 1/a^b
 
  • #8
You can always raise both sides to the same power.
In fact this is exactly what you're doing when solving something like
[tex]3 x^2 = 27[/tex]
If you divide out the 3 you get
[tex]x^2 = 9[/tex]
and you would take the square root to get x = 3 (or - 3 of course). But you can also see it as raising both sides to the power 1/2:
[tex](x^2)^{1/2} = x^{2 \times 1/2} = x \quad = \quad 9^{1/2} = \sqrt{9}[/tex]
where the last equality is just a change of notation, so you see that [tex]\sqrt{x} = x^{1/2}[/tex].

But the raising-both-sides-to-a-power-trick works even in the case of negative and fractional exponents.
 

1. What is a negative exponent?

A negative exponent is a way of writing fractions or powers with a negative number in the exponent. It indicates that the base number should be divided by itself a certain number of times. For example, 2-3 is the same as 1/(23) or 1/8.

2. How do you solve for a negative exponent?

To solve for a negative exponent, you can use the rule: x-n = 1/(xn). This means that you can move the base number from the denominator to the numerator or vice versa, and change the sign of the exponent to positive. For example, 3-2 can be rewritten as 1/(32) or 1/9.

3. What happens when there is a negative exponent in the denominator?

If there is a negative exponent in the denominator, you can use the same rule as above, but with an additional step. First, you can move the base number to the numerator and change the sign of the exponent. Then, you can also flip the fraction so that the negative exponent becomes positive. For example, 1/(5-2) can be rewritten as 52 or 25.

4. Can negative exponents be used with any base number?

Yes, negative exponents can be used with any base number. However, it is important to note that the base number cannot be 0, as any number raised to the power of 0 is equal to 1. Additionally, the base number cannot be negative if the exponent is an odd number, as a negative number raised to an odd power will result in a negative value.

5. How do negative exponents relate to scientific notation?

Negative exponents are often used in scientific notation to represent very small numbers. In scientific notation, a number is written as a decimal between 1 and 10, multiplied by a power of 10. For example, 0.000004 can be rewritten as 4 x 10-6, where the negative exponent indicates that the decimal point needs to be moved 6 places to the left to get the original number.

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