Worksheet: Fractional Exponents

In this worksheet, we will practice simplifying fractional indices.

Q1:

Simplify 3 6 4 × 1 3 6 × ( 1 6 ) .

Q2:

Express 𝑥 in the form 𝑎 .

  • A 𝑥
  • B 𝑥
  • C 𝑥
  • D 𝑥

Q3:

Evaluate 1 2 5 3 4 3 2 3 .

  • A 2 5 7
  • B 5 7
  • C 5 4 9
  • D 2 5 4 9
  • E 7 2 5

Q4:

Evaluate 2 4 3 3 5 .

Q5:

When exponents are written as decimals, they can be converted to fractions:

This makes it easier to write the expressions in radical form:

Now, write 7 2 . 6 in radical form.

  • A 7 1 3 5
  • B 7 1 3 5
  • C 5 7 1 3
  • D 7 5 1 3
  • E 5 1 3 7

Q6:

Write 1 1 3 . 8 in radical form.

  • A 1 9 5 1 1
  • B 1 1 1 9 5
  • C 1 1 1 9 5
  • D 1 1 5 1 9
  • E 1 1 5 1 9

Q7:

Evaluate 1 0 0 0 0 0 .

Q8:

Evaluate 3 1 2 5 3 5 .

Q9:

3 2 = 3 2 = 3 2 × 3 2 × 3 2 = 3 2 = 2 = 8 3 5 1 5 1 5 1 5 1 5 1 5 1 5 5 + + 3 3 .

In general, 𝑎 𝑥 𝑦 means the 𝑥 th exponent of the 𝑦 th root of 𝑎 . So, 3 2 3 5 means “the fifth root of 32, cubed”.

Evaluate 1 6 3 4 .

Q10:

Evaluate 2 5 6 1 4 .

Q11:

8 1 = 8 1 = 8 1 = 2 7 0 . 7 5 3 3 3 4 4 = ( 3 ) .

In this example we could see that 0.75 is equivalent to 3 4 so we need to calculate the cube of the fourth root of 81.

Evaluate 1 6 0 . 7 5 .

Q12:

2 1 6 = 2 1 6 = 2 1 6 = 2 1 6 × 2 1 6 × 2 1 6 = 2 1 6 × 2 1 6 × 2 1 6 1 + + 1 3 1 3 1 3 1 3 1 3 1 3 3 3 3 .

This is an example of using rational exponents to represent radicals. So, 2 1 6 1 3 means “the cube root of 216” or 3 2 1 6 .

Evaluate 2 7 1 3 .

Q13:

Evaluate 2 4 3 0 . 6 .

Q14:

Evaluate 6 4 1 3 .

Q15:

Evaluate 2 5 6 1 8 .

Q16:

Completely simplify 5 . 0 6 2 5 0 . 7 5 .

  • A 2 7 2
  • B 3 2
  • C 3 8
  • D 2 7 8
  • E 8 1 8

Q17:

Evaluate 1 0 0 0 0 0 .

Q18:

Evaluate 6 4 1 2 .

Q19:

Evaluate 3 2 2 5 .

Q20:

Evaluate ( 0 . 0 6 2 5 ) 0 . 2 5 .

  • A 5 1 6
  • B 1 6 4
  • C 1 4
  • D 1 2
  • E 1 8 0

Q21:

4 9 = 4 9 = 4 9 = 4 9 = 4 9 1 ( ) × 2 2 2 1 2 1 2 . This is an example of using rational exponents to represent radicals. So, 4 9 1 2 means “the square root of 49” or 4 9 .

Evaluate 2 5 1 2 .

Q22:

𝑎 𝑏 𝑐 𝑑 means the 𝑐 th exponent of the 𝑑 th root of 𝑎 𝑏 . In the example shown, we can evaluate the numerator and denominator separately.

8 2 7 = 8 2 7 = 8 2 7 = 2 3 = 4 9 2 3 2 3 2 3 3 3 2 2 2 2 .

Evaluate 3 2 2 4 3 3 5 .

  • A 2 2 7
  • B 8 2 4 3
  • C 8 3
  • D 8 2 7
  • E 2 3

Q23:

4 9 = 4 9 = 4 9 × 4 9 = 4 9 × 4 9 1 + 1 2 1 2 1 2 1 2 . This is an example of using rational exponents to represent radicals. So, 4 9 1 2 means “the square root of 49” or 4 9 .

Evaluate 1 6 1 2 .

Q24:

Which of the following is equal to 9?

  • A 8 . 5 1 2
  • B 1 8 1 2
  • C 9 2 1 2
  • D 8 1 1 2
  • E 9 1 2

Q25:

Which of the following expressions is equivalent to 1 3 × 1 3 × 1 3 1 9 2 9 3 9 ?

  • A 1 3 6 9
  • B 1 3 2 3
  • C 3 9 9 6
  • D 1 3 3 2
  • E 3 9 6 9

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