EXERCISE 3 (a) Rewrite the following using the exponential laws. Leave your answers in expone form. (1) $$ (23 \cdot 29)^{4} $$ (2) $$ \left(2011^{5} ight)^{2} $$ (3) $$ (2011)^{5} \cdot(2011)^{5} $$ (4) $$ \left(13^{10} imes 11^{6} ight)^{3} $$ (5) $$ \left(-101^{7} ight)^{2} $$ (6) $$ \left(-101^{7} ight)^{5} $$ b) Simplify: (1) $$ \quad\left(x^{4} ight)^{5} $$ (2) $$ (x y)^{8} $$ (3) $$ \left(t^{3} ight)^{3} $$ (4) $$ (m n)^{4} $$ (5) $$ \quad\left(a^{3} b^{5} ight)^{4} $$ (6) $$ \left(p^{0} q^{3} ight)^{2} $$ (7) $$ \left(3 x^{3} ight)^{2} $$ (8) $$ \left(2 a^{3} ight)^{0} $$ (9) $$ \left(3 a^{2} b^{3} ight)^{2} $$ (10) $$ \quad\left(-x^{2} ight)^{3} $$ (11) $$ \left(-x^{3} ight)^{2} $$ (12) $$ \left[(-x)^{2} ight]^{3} $$ (13) $$ \left(-2 x^{2} y^{3} ight)^{2} $$ (14) $$ \left(-3 p^{2} q^{4} ight)^{3} $$ (15) $$ \left(-2^{100} imes-x^{2} imes(-y) ight. $$ Determine the values of $$ A $$ and $$ B $$ in each of the following: (1) $$ \quad\left(x^{2} y^{\mathrm{A}} ight)^{3}=x^{\mathrm{B}} y^{15} $$ (2) $$ \quad\left(-2 x^{\mathrm{A}} ight)^{3}=\mathrm{B} x^{9} $$

Question

EXERCISE 3 (a) Rewrite the following using the exponential laws. Leave your answers in expone form. (1) $$ (23 \cdot 29)^{4} $$ (2) $$ \left(2011^{5}ight)^{2} $$ (3) $$ (2011)^{5} \cdot(2011)^{5} $$ (4) $$ \left(13^{10} 	imes 11^{6}ight)^{3} $$ (5) $$ \left(-101^{7}ight)^{2} $$ (6) $$ \left(-101^{7}ight)^{5} $$ b) Simplify: (1) $$ \quad\left(x^{4}ight)^{5} $$ (2) $$ (x y)^{8} $$ (3) $$ \left(t^{3}ight)^{3} $$ (4) $$ (m n)^{4} $$ (5) $$ \quad\left(a^{3} b^{5}ight)^{4} $$ (6) $$ \left(p^{0} q^{3}ight)^{2} $$ (7) $$ \left(3 x^{3}ight)^{2} $$ (8) $$ \left(2 a^{3}ight)^{0} $$ (9) $$ \left(3 a^{2} b^{3}ight)^{2} $$ (10) $$ \quad\left(-x^{2}ight)^{3} $$ (11) $$ \left(-x^{3}ight)^{2} $$ (12) $$ \left[(-x)^{2}ight]^{3} $$ (13) $$ \left(-2 x^{2} y^{3}ight)^{2} $$ (14) $$ \left(-3 p^{2} q^{4}ight)^{3} $$ (15) $$ \left(-2^{100} 	imes-x^{2} 	imes(-y)ight. $$ Determine the values of $$ A $$ and $$ B $$ in each of the following: (1) $$ \quad\left(x^{2} y^{\mathrm{A}}ight)^{3}=x^{\mathrm{B}} y^{15} $$ (2) $$ \quad\left(-2 x^{\mathrm{A}}ight)^{3}=\mathrm{B} x^{9} $$

UpStudy AI · Accepted Answer

Let's solve the problems step by step.

### Part (a): Rewrite using exponential laws

1. **For $$ (23 \cdot 29)^{4} $$**:
   $$
   (23 \cdot 29)^{4} = 23^{4} \cdot 29^{4}
   $$

2. **For $$ \left(2011^{5}\right)^{2} $$**:
   $$
   \left(2011^{5}\right)^{2} = 2011^{5 \cdot 2} = 2011^{10}
   $$

3. **For $$ (2011)^{5} \cdot(2011)^{5} $$**:
   $$
   (2011)^{5} \cdot(2011)^{5} = 2011^{5 + 5} = 2011^{10}
   $$

4. **For $$ \left(13^{10} \times 11^{6}\right)^{3} $$**:
   $$
   \left(13^{10} \times 11^{6}\right)^{3} = 13^{10 \cdot 3} \times 11^{6 \cdot 3} = 13^{30} \times 11^{18}
   $$

5. **For $$ \left(-101^{7}\right)^{2} $$**:
   $$
   \left(-101^{7}\right)^{2} = (-1)^{2} \cdot (101^{7})^{2} = 1 \cdot 101^{14} = 101^{14}
   $$

6. **For $$ \left(-101^{7}\right)^{5} $$**:
   $$
   \left(-101^{7}\right)^{5} = (-1)^{5} \cdot (101^{7})^{5} = -1 \cdot 101^{35} = -101^{35}
   $$

### Part (b): Simplify

1. **For $$ \left(x^{4}\right)^{5} $$**:
   $$
   \left(x^{4}\right)^{5} = x^{4 \cdot 5} = x^{20}
   $$

2. **For $$ (x y)^{8} $$**:
   $$
   (x y)^{8} = x^{8} \cdot y^{8}
   $$

3. **For $$ \left(t^{3}\right)^{3} $$**:
   $$
   \left(t^{3}\right)^{3} = t^{3 \cdot 3} = t^{9}
   $$

4. **For $$ (m n)^{4} $$**:
   $$
   (m n)^{4} = m^{4} \cdot n^{4}
   $$

5. **For $$ \left(a^{3} b^{5}\right)^{4} $$**:
   $$
   \left(a^{3} b^{5}\right)^{4} = a^{3 \cdot 4} \cdot b^{5 \cdot 4} = a^{12} \cdot b^{20}
   $$

6. **For $$ \left(p^{0} q^{3}\right)^{2} $$**:
   $$
   \left(p^{0} q^{3}\right)^{2} = p^{0 \cdot 2} \cdot q^{3 \cdot 2} = p^{0} \cdot q^{6} = q^{6}
   $$

7. **For $$ \left(3 x^{3}\right)^{2} $$**:
   $$
   \left(3 x^{3}\right)^{2} = 3^{2} \cdot (x^{3})^{2} = 9 \cdot x^{6} = 9x^{6}
   $$

8. **For $$ \left(2 a^{3}\right)^{0} $$**:
   $$
   \left(2 a^{3}\right)^{0} = 1
   $$

9. **For $$ \left(3 a^{2} b^{3}\right)^{2} $$**:
   $$
   \left(3 a^{2} b^{3}\right)^{2} = 3^{2} \cdot (a^{2})^{2} \cdot (b^{3})^{2} = 9 a^{4} b^{6}
   $$

10. **For $$ \left(-x^{2}\right)^{3} $$**:
    $$
    \left(-x^{2}\right)^{3} = (-1)^{3} \cdot (x^{2})^{3} = -1 \cdot x^{6} = -x^{6}
    $$

11. **For $$ \left(-x^{3}\right)^{2} $$**:
    $$
    \left(-x^{3}\right)^{2} = (-1)^{2} \cdot (x^{3})^{2} = 1 \cdot x^{6} = x^{6}
    $$

12. **For $$ \left[(-x)^{2}\right]^{3} $$**:
    $$
    \left[(-x)^{2}\right]^{3} = (-x)^{2 \cdot 3} = (-x)^{6} = x^{6}
    $$

13. **For $$ \left(-2 x^{2} y^{3}\right)^{2} $$**:
    $$
    \left(-2 x^{2} y^{3}\right)^{2} = (-2)^{2} \cdot (x^{2})^{2} \cdot (y^{3})^{2} = 4 x^{4} y^{6}
    $$

14. **For $$ \left(-3 p^{2} q^{4}\right)^{3} $$**:
    $$
    \left(-3 p^{2} q^{4}\right)^{3} = (-3)^{3} \cdot (p^{2})^{3} \cdot (q^{4})^{3} = -27 p^{6} q^{12}
    $$

15. **For $$ \left(-2^{100} \times -x^{2} \times (-y)\right) $$**:
    $$
    \left(-2^{100} \times -x^{2} \times (-y)\right) = -2^{100} \cdot -1 \cdot -1 \cdot x^{2} \cdot y = -2^{100} x^{2} y
    $$

### Determine the values of $$ A $$ and $$ B $$

1.
1. **For $$ \left(x^{2} y^{\mathrm{A}}\right)^{3} = x^{\mathrm{B}} y^{15} $$**: $$ \left(x^{2} y^{\mathrm{A}}\right)^{3} = x^{2 \cdot 3} \cdot y^{\mathrm{A} \cdot 3} = x^{6} y^{3\mathrm{A}} $$ Comparing with $$ x^{\mathrm{B}} y^{15} $$, we get: $$ 6 = \mathrm{B} \quad \text{and} \quad 3\mathrm{A} = 15 \implies \mathrm{A} = 5 $$ So, $$ \mathrm{A} = 5 $$ and $$ \mathrm{B} = 6 $$. 2. **For $$ \left(-2 x^{\mathrm{A}}\right)^{3} = \mathrm{B} x^{9} $$**: $$ \left(-2 x^{\mathrm{A}}\right)^{3} = (-2)^{3} \cdot (x^{\mathrm{A}})^{3} = -8 x^{3\mathrm{A}} $$ Comparing with $$ \mathrm{B} x^{9} $$, we get: $$ -8 x^{3\mathrm{A}} = \mathrm{B} x^{9} \implies 3\mathrm{A} = 9 \implies \mathrm{A} = 3 $$ And $$ \mathrm{B} = -8 $$. So, $$ \mathrm{A} = 3 $$ and $$ \mathrm{B} = -8 $$.

EXERCISE 3
(a) Rewrite the following using the exponential laws. Leave your answers in expone form.
(1)
(2)
(3)
(4)
(5)
(6)
b) Simplify:
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)

Determine the values of and in each of the following:
(1)
(2)

Solución de inteligencia artificial de Upstudy

Responder

Solución

Part (a): Rewrite using exponential laws

Part (b): Simplify

Determine the values of and

The Deep Dive

a) Rewrite the following using exponential laws:

b) Simplify:

Determine the values of and :

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EXERCISE 3 (a) Rewrite the following using the exponential laws. Leave your answers in expone form. (1) (2) (3) (4) (5) (6) b) Simplify: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) Determine the values of and in each of the following: (1) (2)