Question

note: just as we have implied coefficients and exponents of 1 (for example $1x^1$ is normally written as $x$), we also have an implied index! if you ever see a radical without a stated index, it is implied to be ______!
directions: convert each expression into a radical.
square roots:
$a^{\frac{1}{2}}$
cube roots:
$a^{\frac{1}{3}}$
$n^{th}$ roots:
$a^{\frac{1}{n}}$
directions: convert each expression from exponential to radical form, or vice versa.

1. $49^{\frac{1}{2}}$
2. $\sqrt{12}$
3. $6x^{\frac{1}{2}}$
4. $\sqrt5{(abc)^4}$
5. $7\sqrt4{x^3}$
6. $10^{\frac{1}{2}}$

Explanation:

Step1: Fill implied index blank

A radical without an index has an implied index of 2 (square root).

Step2: Convert 1 to radical form

Use square root rule $a^\frac{1}{2}=\sqrt{a}$
$49^\frac{1}{2}=\sqrt{49}$

Step3: Convert 2 to exponential form

Use square root rule $\sqrt{a}=a^\frac{1}{2}$
$\sqrt{12}=12^\frac{1}{2}$

Step4: Convert 3 to radical form

Use square root rule $x^\frac{1}{2}=\sqrt{x}$
$6x^\frac{1}{2}=6\sqrt{x}$

Step5: Convert 4 to exponential form

Use nth root rule $\sqrt[n]{a^m}=a^\frac{m}{n}$
$\sqrt[5]{(abc)^4}=(abc)^\frac{4}{5}$

Step6: Convert 5 to exponential form

Use nth root rule $\sqrt[n]{a^m}=a^\frac{m}{n}$
$7\sqrt[4]{x^3}=7x^\frac{3}{4}$

Step7: Convert 6 to radical form

Use square root rule $a^\frac{1}{2}=\sqrt{a}$
$10^\frac{1}{2}=\sqrt{10}$

Answer:

Implied index blank: $\boldsymbol{2}$

1. $\sqrt{49}$
2. $12^\frac{1}{2}$
3. $6\sqrt{x}$
4. $(abc)^\frac{4}{5}$
5. $7x^\frac{3}{4}$
6. $\sqrt{10}$