Solved (ODD NUMBERS ONLY) Powers of Monomials & | Chegg.com - Free Printable

Let’s solve each problem step by step. We’re simplifying monomials using exponent rules:

Key Rules to Remember:
- When you raise a power to another power, multiply the exponents: $(x^a)^b = x^{a \cdot b}$
- When you have a product inside parentheses raised to a power, apply the exponent to each factor: $(ab)^n = a^n b^n$
- Don’t forget to handle coefficients (numbers) and signs carefully!

---

1. $(x^4)^2$

Multiply exponents: $4 \cdot 2 = 8$

→ $x^8$

---

2. $(k^5)^9$

Multiply exponents: $5 \cdot 9 = 45$

→ $k^{45}$

---

3. $(m^2 n^7)^3$

Apply exponent to each part:
$m^{2 \cdot 3} = m^6$,
$n^{7 \cdot 3} = n^{21}$

→ $m^6 n^{21}$

---

4. $(2w^3)^6$

Apply exponent to 2 and to $w^3$:
$2^6 = 64$,
$(w^3)^6 = w^{18}$

→ $64w^{18}$

---

5. $(3y^3)^4$

Apply exponent to 3 and to $y^3$:
$3^4 = 81$,
$(y^3)^4 = y^{12}$

→ $81y^{12}$

---

6. $(-4x^2)^2$

Square both -4 and $x^2$:
$(-4)^2 = 16$,
$(x^2)^2 = x^4$

→ $16x^4$

---

7. $(-5t^3)^3$

Cube both -5 and $t^3$:
$(-5)^3 = -125$,
$(t^3)^3 = t^9$

→ $-125t^9$

---

8. $(-a^9 b)^4$

Apply exponent to each part:
$(-1)^4 = 1$, so sign becomes positive
$(a^9)^4 = a^{36}$,
$b^4 = b^4$

→ $a^{36} b^4$

---

9. $\left(\frac{1}{2} x^4\right)^7$

Apply exponent to fraction and variable:
$\left(\frac{1}{2}\right)^7 = \frac{1}{128}$,
$(x^4)^7 = x^{28}$

→ $\frac{1}{128} x^{28}$

---

10. $2(3a^2)^3$

First simplify inside: $(3a^2)^3 = 3^3 \cdot (a^2)^3 = 27a^6$
Then multiply by 2: $2 \cdot 27a^6 = 54a^6$

→ $54a^6$

---

11. $\frac{1}{2}(4k^5)^2$

First: $(4k^5)^2 = 4^2 \cdot (k^5)^2 = 16k^{10}$
Then multiply by $\frac{1}{2}$: $\frac{1}{2} \cdot 16k^{10} = 8k^{10}$

→ $8k^{10}$

---

12. $-6\left(\frac{2}{3} p^8\right)^3$

First: $\left(\frac{2}{3}\right)^3 = \frac{8}{27}$,
$(p^8)^3 = p^{24}$
So inside: $\frac{8}{27} p^{24}$
Now multiply by -6: $-6 \cdot \frac{8}{27} p^{24} = -\frac{48}{27} p^{24}$

Simplify fraction: divide numerator and denominator by 3 → $-\frac{16}{9} p^{24}$

→ $-\frac{16}{9} p^{24}$

---

Final Answer:
1. $x^8$
2. $k^{45}$
3. $m^6 n^{21}$
4. $64w^{18}$
5. $81y^{12}$
6. $16x^4$
7. $-125t^9$
8. $a^{36} b^4$
9. $\frac{1}{128} x^{28}$
10. $54a^6$
11. $8k^{10}$
12. $-\frac{16}{9} p^{24}$