1. **Stating the problem:** We want to understand the laws of indices (exponents) and how they relate to repeated multiplication, which can be seen as a form of repeated addition in multiplication. 2. **Expanded form example:** $$2 \times 2 \times 2 = 2^3$$ This means multiplying 2 by itself 3 times. 3. **General form:** $$a^n = \underbrace{a \times a \times \cdots \times a}_{n \text{ times}}$$ where $a$ is the base and $n$ is the exponent. 4. **Multiplying powers with different bases but related:** $$2^3 \times 4^2$$ Since $4 = 2^2$, rewrite: $$2^3 \times (2^2)^2 = 2^3 \times 2^{4} = 2^{3+4} = 2^7$$ Note: The user wrote $2^3 \times 4^2 = 8^5$ which is incorrect. Correct simplification is $2^7$. 5. **Fractional exponents and roots:** $$8^{\frac{1}{3}} = \sqrt[3]{8}$$ Since $8 = 2 \times 2 \times 2 = 2^3$, the cube root is: $$\sqrt[3]{8} = 2$$ 6. **Another example:** $$27^{\frac{1}{3}} = \sqrt[3]{27}$$ Since $27 = 3 \times 3 \times 3 = 3^3$, the cube root is: $$\sqrt[3]{27} = 3$$ 7. **Summary:** - Exponents represent repeated multiplication. - Multiplying powers with the same base adds exponents: $$a^m \times a^n = a^{m+n}$$ - Fractional exponents represent roots: $$a^{\frac{1}{n}} = \sqrt[n]{a}$$ This shows how laws of indices relate to repeated multiplication and roots.