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Here are the Laws of Exponents explained in English:

The Laws of Exponents are a set of rules that govern how to handle exponents in algebraic expressions. These rules simplify calculations and help in solving equations.

Here are some fundamental rules:

• Product Rule: When multiplying powers with the same base, add the exponents.

  $$x^m \cdot x^n = x^{m+n}$$

  Example: $2^3 \cdot 2^4 = 2^{3+4} = 2^7$

• Quotient Rule: When dividing powers with the same base, subtract the exponents.

  $$\frac{x^m}{x^n} = x^{m-n}$$

  Example: $\frac{5^6}{5^2} = 5^{6-2} = 5^4$

• Power of a Power Rule: When raising a power to another power, multiply the exponents.

  $$(x^m)^n = x^{mn}$$

  Example: $(3^2)^3 = 3^{2 \cdot 3} = 3^6$

• Power of a Product Rule: When raising a product to a power, raise each factor to that power.

  $$(xy)^n = x^n y^n$$

  Example: $(2 \cdot 5)^3 = 2^3 \cdot 5^3$

• Power of a Quotient Rule: When raising a quotient to a power, raise both the numerator and the denominator to that power.

  $$\left(\frac{x}{y}\right)^n = \frac{x^n}{y^n}$$

  Example: $\left(\frac{4}{7}\right)^2 = \frac{4^2}{7^2}$

• Zero Exponent Rule: Any non-zero number raised to the power of zero is 1.

  $$x^0 = 1 \quad (\text{where } x \neq 0)$$

  Example: $10^0 = 1$

• Negative Exponent Rule: A negative exponent means the reciprocal of the base raised to the positive exponent.

  $$x^{-n} = \frac{1}{x^n}$$

  Example: $4^{-2} = \frac{1}{4^2} = \frac{1}{16}$

• Fractional Exponent Rule: Fractional exponents represent roots.

  $$x^{1/n} = \sqrt[n]{x}$$ $$x^{m/n} = \sqrt[n]{x^m} = (\sqrt[n]{x})^m$$

  Example: $9^{1/2} = \sqrt{9} = 3$ Example: $8^{2/3} = \sqrt[3]{8^2} = \sqrt[3]{64} = 4$