Understanding and Evaluating Exponents

• An exponent tells you how many times to use the base as a factor.
• A base is the number that is multiplied by itself when raised to an exponent.
• Exponents are a way to express repeated multiplication in a concise format.
• For example, (5^3) indicates that 5 is multiplied by itself three times: (5 \times 5 \times 5).
• The exponent is also known as the power.
• Evaluating exponents involves finding the standard form of a number raised to a power.

Evaluating Exponents

• To evaluate an exponent, perform the multiplication that the exponent indicates.
• For example, to evaluate (2^4), multiply 2 by itself four times: (2 \times 2 \times 2 \times 2 = 16).
• When evaluating exponents with negative bases, pay close attention to the exponent.
• A negative base raised to an even exponent results in a positive number.
• For example, ((-3)^2 = (-3) \times (-3) = 9).
• A negative base raised to an odd exponent results in a negative number.
• For example, ((-3)^3 = (-3) \times (-3) \times (-3) = -27).
• Any non-zero number raised to the power of 0 is 1. For example, (7^0 = 1).

Laws of Exponents

• The laws of exponents provide rules for simplifying expressions involving exponents. These rules make it easier to work with powers.

Product of Powers Rule

• When multiplying powers with the same base, add the exponents.
• Expressed as: (a^m \times a^n = a^{m+n}), where 'a' is the base, and 'm' and 'n' are the exponents.
• For example, (2^3 \times 2^4 = 2^{3+4} = 2^7 = 128).

Quotient of Powers Rule

• When dividing powers with the same base, subtract the exponents.
• Expressed as: (a^m \div a^n = a^{m-n}), where 'a' is the base, and 'm' and 'n' are the exponents.
• For example, (3^5 \div 3^2 = 3^{5-2} = 3^3 = 27).

Power of a Power Rule

• When raising a power to another power, multiply the exponents.
• Expressed as: ((a^m)^n = a^{m \times n}), where 'a' is the base, and 'm' and 'n' are the exponents.
• For example, ((4^2)^3 = 4^{2 \times 3} = 4^6 = 4096).

Power of a Product Rule

• When raising a product to a power, distribute the exponent to each factor in the product.
• Expressed as: ((ab)^n = a^n \times b^n), where 'a' and 'b' are the bases, and 'n' is the exponent.
• For example, ((2x)^3 = 2^3 \times x^3 = 8x^3).

Power of a Quotient Rule

• When raising a quotient to a power, distribute the exponent to both the numerator and the denominator.
• Expressed as: ((a/b)^n = a^n / b^n), where 'a' and 'b' are the bases, and 'n' is the exponent.
• For example, ((3/y)^2 = 3^2 / y^2 = 9 / y^2).

Zero Exponent Rule

• Any non-zero number raised to the power of 0 is 1.
• Expressed as: (a^0 = 1), where (a \neq 0).
• For example, (5^0 = 1).

Negative Exponent Rule

• A negative exponent indicates that the base should be taken as a reciprocal.
• Expressed as: (a^{-n} = 1 / a^n), where 'a' is the base, and 'n' is the exponent.
• For example, (2^{-3} = 1 / 2^3 = 1 / 8).
• Also, ((a/b)^{-n} = (b/a)^n). For example, ((2/3)^{-2} = (3/2)^2 = 9/4).

Applying the Laws of Exponents

• Simplify expressions by applying one or more of the exponent laws.
• When simplifying, it's important to follow the order of operations (PEMDAS/BODMAS).
• For expressions with multiple operations, simplify inside parentheses first, then apply exponents, followed by multiplication and division, and finally addition and subtraction.
• Keep simplifying until each base appears only once and all exponents are positive.
• Simplify ((3x^2y)^2 \times (2x^{-1}y^3))
• Apply power of a product rule: (9x^4y^2 \times (2x^{-1}y^3))
• Apply commutative property: (9 \times 2 \times x^4 \times x^{-1} \times y^2 \times y^3)
• Then, (18 x^{4-1} y^{2+3})
• Finally, (18 x^3 y^5)