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Common Core Standards: Math

High School: Number and Quantity

The Real Number System HSN-RN.A.2

2. Rewrite expressions using radical and rational exponents using the properties of exponents.

In the previous standard, we established some rules about fractional exponents. Here's the gist of those rules:

  1. We're allowed to have exponents that are fractions. It's really okay.
  2. The denominator of the fraction is the root.
  3. The numerator of the fraction is the power. 
  4. It doesn't matter which we do first. 

Those four little rules mean that it's easy to evaluate a lot of fractional exponents without the use of a calculator. Your students should know that math—real math with scary things like exponents—can be done without a calculator. Yes, using the calculator function on their phone counts as cheating. A pencil and an eraser, on the other hand, does not.

Now that we know these rules, we can go from radicals to exponents and vice versa. That'll help us turn some pretty ugly-duckling problems into gorgeous-swan answers. Or at least make the transition from duck to swan significantly less painful than ballet dancer to swan.

For example, let's rewrite  without a radical and evaluate. The 3 outside the radical becomes the denominator. So what do we do for a numerator? The numerator is the power to which 64 is raised. Since no power is listed, we know it's a power of 1. (A number that doesn't have an exponent is to the first power.)

In exponential form, our answer is . To evaluate it, we just find the number that, when cubed is 64. That'd be 4.

What if we want to rewrite  in radical form? Well, our numerator is 7 and our denominator is 2. We want the square root of 100 to the seventh power, or . (Where did the 2 go? For square roots, the 2 is assumed. Did we assume 2 much?)

If we raise 10 to the seventh power, we get the very long number 10,000,000. If only that was the number in your bank account, huh?

Drills

  1. Rewrite  in radical form.

    Correct Answer:

    Answer Explanation:

    We use the denominator of the exponent to tell us the root. In this case, it's a 2, so we want the square root. We use the numerator to determine the power, so we want the first power. The square root of 64 raised to the first power is .


  2. Rewrite   without the radical.

    Correct Answer:

    Answer Explanation:

    Radicals have to do with fractional exponents, not negative exponents. That little 4 outside the radical, (called the index just in case you were burning with curiosity) tells you that you want the fourth root. So the 4 becomes the denominator. Since the 81 has no other exponent, it's being raised to the first power. That means the numerator is 1. So 81 is raised to the ¼ power.


  3. Rewrite  in radical form.

    Correct Answer:

    Answer Explanation:

    The denominator of the exponent, 3, gives us the root, or the index. (That's the little number outside the radical.) The numerator gives us the power. We want 27 squared under the cube root. Translate that into mathspeak, and we get .


  4. Evaluate

    Correct Answer:

    64

    Answer Explanation:

    First, rewrite (or at least re-think) the problem in radical form. We want the square root of 16, cubed. The square root of 16 is 4. (No, it's not 8! We want the square root, not half.) Once we get that 4, we want to cube it, and 43 is 64.


  5. Evaluate .

    Correct Answer:

    10,000

    Answer Explanation:

    Once again, we want to re-imagine our problem in radical form. We want the cubed root of 1,000, raised to the fourth power. The cubed root of 1,000 is 10, right? (Because 10 × 10 × 10 = 1,000.) When we raise 10 to the fourth power we get 10,000.


  6. Which of the following is equal to 10?

    Correct Answer:

    Answer Explanation:

    Make sure to count the zeros in this one. We know that  or  is the same as 10, so (A) won't give us the right value. The cube root of 1002 (or 10,000) equals about 21.5, so that's not right, but taking the cube root of 1,000, will give us the right answer.


  7. Which of the following is the same as ?

    Correct Answer:

    Answer Explanation:

    Remember that the numerator is our power and the denominator is our root. Knowing that, we can set up the fraction  as the exponent (not ). Simplifying it, we get  as our power. That means we're looking for either 8 to the cube root or 8 to the power of one third. That's (B).


  8. Which of the following is the same as ?

    Correct Answer:

    8

    Answer Explanation:

    We want to take the fourth root of 16 and raise it to the third power. If we look at our answers, (D) has the incorrect base and (B) takes the square root, not the fourth root. The fourth root is just the square root of the square root. That means we can take the square root of 16 (which is 4) and we'd be left with  (unlike (C), which has 8 as the base). If we calculate it, we'll get 8, which is (A).


  9. Which of the following is the same as ?

    Correct Answer:

    Answer Explanation:

    We can rearrange the number into exponent form, which will give us  . That fraction in the exponent is being complicated on purpose, so let's simplify it to 3 (we know how to divide, don't we?). That rules (B) out automatically, and same with (C). If we translate (A) into exponent mode, we'll get 54, which isn't what we want, so (D) is the right answer.


  10. Which of the following is the same as ?

    Correct Answer:

    67

    Answer Explanation:

    Rearranging this into fraction form, we have . We know that   = 7, so our exponent is only a 7. The answer (D) is wrong because it changes the base entirely, and (B) is wrong because the numerator and denominator in the exponent are switched. The correct answer is (C) because the exponent's denominator is 1, not 2 like it would be if (A) were rewritten.


More standards from High School: Number and Quantity - The Real Number System