Question

Simplify
1.) $\frac{r^5}{r^2}$ 2.) $\frac{3t^5}{6t^{10} }$

3.) $\frac{18a^5b^2}{6a^3b^8}$ 4.) $\frac{xy^2}{5x^3y^5}$

5.) $\frac{-7e^{15}f^4}{21e^3f^8}$ 6.) $\frac{-3hj^4}{21hj}$

Answer 1

Answer:

Certainly! Let me simplify each expression for you:

1. \( \frac{\sqrt{5} \cdot r^2}{2r^5} \) simplifies to \( \frac{1}{2\sqrt{5}r^3} \)

2. \( \frac{3\sqrt[5]{6t}}{10\sqrt[3]{5}} \) simplifies to \( \frac{\sqrt[3]{2t}}{5} \)

3. \( \frac{18a^3b}{5b^2 \cdot 2} \) simplifies to \( \frac{9a^2}{5} \)

4. \( \frac{\sqrt{2} \cdot 5xy^3}{3 \cdot 5^3} \) simplifies to \( \frac{\sqrt{2}xy}{75} \)

5. \( \frac{-7e^4f}{15f^3 \cdot 4} \) simplifies to \( \frac{-7e}{60f^2} \)

6. \( \frac{-3h^2}{4 \cdot 21h^2j} \) simplifies to \( \frac{-1}{28j} \)

Answer 2

Answer:

1. $r^3$
2. $\dfrac{1}{2t^5}$
3. $\dfrac{3a^2}{b^6}$
4. $\dfrac{1}{5xy^3}$
5. $ -\dfrac{e^{12}}{3f^4}$
6. $-\dfrac{1}{7}j^3$

Step-by-step explanation:

The laws of indices are rules that govern the manipulation and simplification of expressions involving exponents. Here are some of the fundamental laws of indices:

• Multiplication: $a^m \times a^n = a^{m+n}$
• Division: $\dfrac{a^m}{a^n} = a^{m-n}$
• Power of a Power: $(a^m)^n = a^{mn}$
• Power of a Product: $(ab)^n = a^n b^n$
• Negative Exponent: $a^{-n} = \dfrac{1}{a^n}$

• Zero Exponent: $a^0 = 1$

Let's simplify the given expressions using these laws:

1. $\dfrac{r^5}{r^2}$

  Using the division law:

$\dfrac{r^5}{r^2} = r^{5-2}\\\\ = r^3$.

2. $\dfrac{3t^5}{6t^{10}}$

  We can simplify by dividing both the coefficient and the exponents:

  $\dfrac{3t^5}{6t^{10}} = \dfrac{1}{2}t^{5-10} \\\\= \dfrac{1}{2}t^{-5}\\\\= \dfrac{1}{2t^5}$.

3. $\dfrac{18a^5b^2}{6a^3b^8}$

 Divide the coefficients and simplify the variables:

  $\dfrac{18a^5b^2}{6a^3b^8} = \dfrac{18}{6} \times \dfrac{a^5}{a^3} \times \dfrac{b^2}{b^8} \\\\= 3a^{5-3}b^{2-8} \\\\= 3a^2b^{-6}\\\\ = \dfrac{3a^2}{b^6}$.

4. $\dfrac{xy^2}{5x^3y^5}$

 Divide the coefficients and simplify the variables:

  $\dfrac{xy^2}{5x^3y^5} = \dfrac{1}{5} \times \dfrac{y^2}{y^5} \times \dfrac{x}{x^3} \\\\= \dfrac{1}{5}y^{2-5}x^{1-3}\\\\ = \dfrac{1}{5}y^{-3}x^{-2}\\\\ = \dfrac{1}{5xy^3}$.

5. $\dfrac{-7e^{15}f^4}{21e^3f^8}$

  Divide the coefficients and simplify the variables:

  $\dfrac{-7e^{15}f^4}{21e^3f^8} = \dfrac{-7}{21} \times \dfrac{e^{15}}{e^3} \times \dfrac{f^4}{f^8} \\\\= -\dfrac{1}{3}e^{15-3}f^{4-8}\\\\ = -\dfrac{1}{3}e^{12}f^{-4}\\\\ = -\dfrac{e^{12}}{3f^4}$.

6. $\dfrac{-3hj^4}{21hj}$

  Divide the coefficients and simplify the variables:

  $\dfrac{-3hj^4}{21hj} = \dfrac{-3}{21} \times \dfrac{j^4}{j} \\\\= -\dfrac{1}{7}j^{4-1} \\\\= -\dfrac{1}{7}j^3$.