Question

all answers need an exponent! no exponents, no points.
guided practice
lets try these together.

1. $(4^{2})^{3}$
2. $(b^{7})^{8}$
3. $sqrt{4^{2}}$
4. $sqrt{x^{3}}$
5. $sqrt3{2^{4}}$
6. $sqrt3{x^{5}}$
7. $sqrt3{s^{2}m^{4}h^{3} \cdot s^{7}m^{2}}$

secret parentheses!
always simplify inside the root before dividing the exponents!
independent practice
try these on your own.

1. $((-2)^{3})^{3}$
2. $(2^{2})^{-2}$
3. $(x^{2})^{-3}$
4. $sqrt{\frac{x^{6}}{x^{4}}}$
5. $sqrt3{a^{2}b^{2}c^{3} \cdot ab}$
6. $(3 \cdot 3^{2})^{2}$
7. $(x^{6})^{5}$
8. $sqrt3{\frac{2^{7}}{2^{3}}}$
9. $sqrt{xy^{2}z^{4} \cdot x^{2}y^{2}}$
10. $sqrt3{(z^{2})^{4}}$
11. $(y^{2})^{39}$
12. $(4^{2})^{3}$
13. $\left(\frac{6^{4}}{6^{2}}\

ight)^{3}$

1. $sqrt{\frac{x^{4}y^{2}}{x^{2}y}}$
2. $sqrt{s^{2}m^{2}h^{2}}$
3. $sqrt3{2^{6}}$
4. $sqrt{3^{5} \cdot 3^{2}}$
5. $((x^{3})^{2})^{2}$
6. $sqrt3{\frac{x^{8}y^{9}}{x^{3}y^{4}}}$
7. $sqrt{(x^{2})^{3}}$

Explanation:

Guided Practice

Step1: Power of a power rule

For $(4^2)^3$, multiply exponents: $2 \times 3 = 6$, so $4^{2 \times 3} = 4^6$

Step2: Power of a power rule

For $(b^7)^8$, multiply exponents: $7 \times 8 = 56$, so $b^{7 \times 8} = b^{56}$

Step3: Root as fractional exponent

$\sqrt{4^2} = (4^2)^{\frac{1}{2}}$, multiply exponents: $2 \times \frac{1}{2} = 1$, so $4^{2 \times \frac{1}{2}} = 4^1$

Step4: Root as fractional exponent

$\sqrt{x^3} = (x^3)^{\frac{1}{2}}$, multiply exponents: $3 \times \frac{1}{2} = \frac{3}{2}$, so $x^{3 \times \frac{1}{2}} = x^{\frac{3}{2}}$

Step5: Root as fractional exponent

$\sqrt[3]{2^4} = (2^4)^{\frac{1}{3}}$, multiply exponents: $4 \times \frac{1}{3} = \frac{4}{3}$, so $2^{4 \times \frac{1}{3}} = 2^{\frac{4}{3}}$

Step6: Root as fractional exponent

$\sqrt[3]{x^5} = (x^5)^{\frac{1}{3}}$, multiply exponents: $5 \times \frac{1}{3} = \frac{5}{3}$, so $x^{5 \times \frac{1}{3}} = x^{\frac{5}{3}}$

Step7: Simplify inside root first

Multiply like bases: $s^{2+1}m^{4+2}h^{3} = s^3m^6h^3$. Apply cube root: $(s^3m^6h^3)^{\frac{1}{3}}$, multiply exponents: $s^{3 \times \frac{1}{3}}m^{6 \times \frac{1}{3}}h^{3 \times \frac{1}{3}} = s^1m^2h^1$

Independent Practice (1-10)

Step1: Power of a power rule

For $((-2)^3)^3$, multiply exponents: $3 \times 3 = 9$, so $(-2)^{3 \times 3} = (-2)^9$

Step2: Power of a power rule

For $(2^2)^{-2}$, multiply exponents: $2 \times (-2) = -4$, so $2^{2 \times (-2)} = 2^{-4}$

Step3: Power of a power rule

For $(x^2)^{-3}$, multiply exponents: $2 \times (-3) = -6$, so $x^{2 \times (-3)} = x^{-6}$

Step4: Simplify inside root first

Subtract exponents: $\frac{x^6}{x^4} = x^{6-4} = x^2$. Apply square root: $(x^2)^{\frac{1}{2}} = x^{2 \times \frac{1}{2}} = x^1$

Step5: Simplify inside root first

Multiply like bases: $a^{2+1}b^{2+1}c^3 = a^3b^3c^3$. Apply cube root: $(a^3b^3c^3)^{\frac{1}{3}} = a^{3 \times \frac{1}{3}}b^{3 \times \frac{1}{3}}c^{3 \times \frac{1}{3}} = a^1b^1c^1$

Step6: Simplify inside parentheses first

$3 \times 3^2 = 3^{1+2} = 3^3$. Apply power of a power: $(3^3)^2 = 3^{3 \times 2} = 3^6$

Step7: Power of a power rule

For $(x^6)^5$, multiply exponents: $6 \times 5 = 30$, so $x^{6 \times 5} = x^{30}$

Step8: Simplify inside root first

Subtract exponents: $\frac{2^7}{2^3} = 2^{7-3} = 2^4$. Apply cube root: $(2^4)^{\frac{1}{3}} = 2^{4 \times \frac{1}{3}} = 2^{\frac{4}{3}}$

Step9: Simplify inside root first

Multiply like bases: $x^{1+2}y^{2+2}z^4 = x^3y^4z^4$. Apply square root: $(x^3y^4z^4)^{\frac{1}{2}} = x^{3 \times \frac{1}{2}}y^{4 \times \frac{1}{2}}z^{4 \times \frac{1}{2}} = x^{\frac{3}{2}}y^2z^2$

Step10: Power of a power first

$(z^2)^4 = z^{8}$. Apply cube root: $(z^8)^{\frac{1}{3}} = z^{8 \times \frac{1}{3}} = z^{\frac{8}{3}}$

Independent Practice (11-20)

Step1: Power of a power rule

For $(y^2)^{39}$, multiply exponents: $2 \times 39 = 78$, so $y^{2 \times 39} = y^{78}$

Step2: Power of a power rule

For $(4^2)^3$, multiply exponents: $2 \times 3 = 6$, so $4^{2 \times 3} = 4^6$

Step3: Simplify inside parentheses first

Subtract exponents: $\frac{6^4}{6^2} = 6^{4-2} = 6^2$. Apply power of a power: $(6^2)^3 = 6^{2 \times 3} = 6^6$

Step4: Simplify inside root first

Subtract exponents: $\frac{x^4y^2}{x^2y} = x^{4-2}y^{2-1} = x^2y^1$. Apply square root: $(x^2y^1)^{\frac{1}{2}} = x^{2 \times \frac{1}{2}}y^{1 \times \frac{1}{2}} = x^1y^{\frac{1}{2}}$

Step5: Apply square root to each term

$\sqrt{s^2m^2h^2} = (s^2m^2h^2)^{\frac{1}{2}} = s^{2 \times \frac{1}{2}}m^{2 \times \frac{1}{2}}h^{2 \times \frac{1}{2}} = s^1m^1h^1$

Step6: Power of a power rule

$\sqrt[3]{2^6} = (2^6)^{\frac{1}{3}}$, multiply exponents: $6 \times \frac{1}{3} = 2$, so $2^{6 \times \frac{1}{3}} = 2^2$

Step7: Simplify inside root first

Add exponents: $3^5 \times 3^2 = 3^{5+2} = 3^7$. Apply square root: $(3^7)^{\frac{1}{2}} = 3^{7 \times \frac{1}{2}} = 3^{\frac{7}{2}}$

Step8: Power of a power rule

For $((x^3)^2)^2$, multiply exponents: $3 \times 2 \times 2 = 12$, so $x^{3 \times 2 \times 2} = x^{12}$

Step9: Simplify inside root first

Subtract exponents: $\frac{x^8y^9}{x^3y^4} = x^{8-3}y^{9-4} = x^5y^5$. Apply cube root: $(x^5y^5)^{\frac{1}{3}} = x^{5 \times \frac{1}{3}}y^{5 \times \frac{1}{3}} = x^{\frac{5}{3}}y^{\frac{5}{3}}$

Step10: Power of a power first

$(x^2)^3 = x^6$. Apply square root: $(x^6)^{\frac{1}{2}} = x^{6 \times \frac{1}{2}} = x^3$

Answer:

Guided Practice:

1. $4^6$
2. $b^{56}$
3. $4^1$
4. $x^{\frac{3}{2}}$
5. $2^{\frac{4}{3}}$
6. $x^{\frac{5}{3}}$
7. $s^{\frac{3}{3}}m^{\frac{6}{3}}h^{\frac{3}{3}} = s^1m^2h^1$

Independent Practice (1-10):

1. $(-2)^9$
2. $2^{-4}$
3. $x^{-6}$
4. $x^1$
5. $a^1b^1c^1$
6. $3^6$
7. $x^{30}$
8. $2^{\frac{4}{3}}$
9. $x^{\frac{3}{2}}y^2z^2$
10. $z^{\frac{8}{3}}$

Independent Practice (11-20):

1. $y^{78}$
2. $4^6$
3. $6^6$
4. $x^1y^{\frac{1}{2}}$
5. $s^1m^1h^1$
6. $2^2$
7. $3^{\frac{7}{2}}$
8. $x^{12}$
9. $x^{\frac{5}{3}}y^{\frac{5}{3}}$
10. $x^3$