Simplify the following withou using a calculator. $$ \begin{array}{lll} ext { (a) } \ \begin{array}{ll} ext { (d) } \frac{2 x^{2} y \cdot 4 x^{4} y^{3}}{6 x^{6} y^{6}} & ext { (b) }\left(-2 x^{3} y^{2} ight)^{3}\end{array} & ext { (c) }\left(3^{3} ight)^{2} \cdot 3^{2}\end{array} $$ $$ \begin{array}{ll} ext { (g) } \frac{\left(-3 a^{2} b^{4} ight)^{2}}{3\left(a^{2} b ight)^{3}} & ext { (e) }\left(\frac{-12 x^{3}}{15 y^{-2}} ight)^{2}\end{array} $$ (1) $$ )\left(\frac{10 y^{4}}{4 y^{6}} ight)^{-2} $$ i) $$ \frac{x+y}{x^{-1}+y^{-1}} $$

Question

Simplify the following withou using a calculator. $$ \begin{array}{lll}	ext { (a) } \ \begin{array}{ll}	ext { (d) } \frac{2 x^{2} y \cdot 4 x^{4} y^{3}}{6 x^{6} y^{6}} & 	ext { (b) }\left(-2 x^{3} y^{2}ight)^{3}\end{array} & 	ext { (c) }\left(3^{3}ight)^{2} \cdot 3^{2}\end{array} $$ $$ \begin{array}{ll}	ext { (g) } \frac{\left(-3 a^{2} b^{4}ight)^{2}}{3\left(a^{2} bight)^{3}} & 	ext { (e) }\left(\frac{-12 x^{3}}{15 y^{-2}}ight)^{2}\end{array} $$ (1) $$ )\left(\frac{10 y^{4}}{4 y^{6}}ight)^{-2} $$ i) $$ \frac{x+y}{x^{-1}+y^{-1}} $$

UpStudy AI · Accepted Answer

Certainly! Let's simplify each of the given expressions step by step without using a calculator.

---

### **(b) $$ \left(-2 x^{3} y^{2}\right)^{3} $$**

**Solution:**
Apply the exponent to each factor inside the parentheses:
$$
(-2)^3 \cdot (x^3)^3 \cdot (y^2)^3 = -8 \cdot x^9 \cdot y^6
$$

**Simplified Form:**
$$
-8x^{9}y^{6}
$$

---

### **(c) $$ \left(3^{3}\right)^{2} \cdot 3^{2} $$**

**Solution:**
First, simplify the exponents:
$$
(3^3)^2 = 3^{3 \cdot 2} = 3^6
$$
Then multiply by $$3^2$$:
$$
3^6 \cdot 3^2 = 3^{6+2} = 3^8
$$
Calculate $$3^8$$ if a numerical answer is desired:
$$
3^8 = 6561
$$

**Simplified Form:**
$$
3^{8} \quad \text{or} \quad 6561
$$

---

### **(d) $$ \frac{2 x^{2} y \cdot 4 x^{4} y^{3}}{6 x^{6} y^{6}} $$**

**Solution:**
Multiply the numerators and then simplify:
$$
\frac{2 \cdot 4 \cdot x^{2+4} \cdot y^{1+3}}{6 x^6 y^6} = \frac{8x^6 y^4}{6x^6 y^6}
$$
Simplify the coefficients and exponents:
$$
\frac{8}{6} \cdot x^{6-6} \cdot y^{4-6} = \frac{4}{3} \cdot y^{-2} = \frac{4}{3y^2}
$$

**Simplified Form:**
$$
\frac{4}{3y^{2}}
$$

---

### **(e) $$ \left(\frac{-12 x^{3}}{15 y^{-2}}\right)^{2} $$**

**Solution:**
Simplify inside the parentheses first:
$$
\frac{-12x^3}{15y^{-2}} = \frac{-12x^3 \cdot y^2}{15} = -\frac{4}{5}x^3 y^2
$$
Now, square each component:
$$
\left(-\frac{4}{5}\right)^2 \cdot (x^3)^2 \cdot (y^2)^2 = \frac{16}{25}x^6 y^4
$$

**Simplified Form:**
$$
\frac{16x^{6}y^{4}}{25}
$$

---

### **(g) $$ \frac{\left(-3 a^{2} b^{4}\right)^{2}}{3\left(a^{2} b\right)^{3}} $$**

**Solution:**
Simplify the numerator and the denominator separately:
$$
(-3a^2b^4)^2 = 9a^4b^8
$$
$$
3(a^2b)^3 = 3a^6b^3
$$
Now, divide the two:
$$
\frac{9a^4b^8}{3a^6b^3} = 3a^{4-6}b^{8-3} = 3a^{-2}b^5 = \frac{3b^5}{a^2}
$$

**Simplified Form:**
$$
\frac{3b^{5}}{a^{2}}
$$

---

### **(1) $$ \left(\frac{10 y^{4}}{4 y^{6}}\right)^{-2} $$**

**Solution:**
Simplify inside the parentheses first:
$$
\frac{10y^4}{4y^6} = \frac{5}{2}y^{-2}
$$
Now, apply the exponent:
$$
\left(\frac{5}{2}y^{-2}\right)^{-2} = \left(\frac{5}{2}\right)^{-2} \cdot y^{4} = \frac{4}{25}y^4
$$

**Simplified Form:**
$$
\frac{4y^{4}}{25}
$$

---

### **(i) $$ \frac{x+y}{x^{-1} + y^{-1}} $$**

**Solution:**
Simplify the denominator:
$$
x^{-1} + y^{-1} = \frac{1}{x} + \frac{1}{y} = \frac{y + x}{xy}
$$
Now, divide the numerator by this result:
$$
\frac{x + y}{\frac{x + y}{xy}} = (x + y) \cdot \frac{xy}{x + y} = xy
$$

**Simplified Form:**
$$
xy
$$

---

**Summary of Simplified Forms:**

- **(b):** $$ -8x^{9}y^{6} $$
- **(c):** $$ 3^{8} $$ or $$ 6561 $$
- **(d):** $$ \dfrac{4}{3y^{2}} $$
- **(e):** $$ \dfrac{16x^{6}y^{4}}{25} $$
- **(g):** $$ \dfrac{3b^{5}}{a^{2}} $$
- **(1):** $$ \dfrac{4y^{4}}{25} $$
- **(i):** $$ xy $$
Here are the simplified forms of the given expressions: - **(b):** $$ -8x^{9}y^{6} $$ - **(c):** $$ 3^{8} $$ or $$ 6561 $$ - **(d):** $$ \dfrac{4}{3y^{2}} $$ - **(e):** $$ \dfrac{16x^{6}y^{4}}{25} $$ - **(g):** $$ \dfrac{3b^{5}}{a^{2}} $$ - **(1):** $$ \dfrac{4y^{4}}{25} $$ - **(i):** $$ xy $$

Simplify the following withou using a calculator.

(1)
i)

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Simplify the following withou using a calculator. (1) i)