Multiply by 9

Learning Objectives Express the number 9 as a power with a base of 3. Apply the product of powers property to multiply expressions with rational exponents by 9. Simplify expressions involving rational exponents where the base is a power of 3. Convert numbers like 27, 81, and 243 into base 3 to simplify multiplication by 9. Solve equations involving rational exponents and multiplication by 9. Evaluate numerical expressions containing rational exponents and the factor 9 without a calculator. How can the simple act of multiplying by 9 unlock the secrets of advanced algebraic expressions? 🤔 Let's find out! This tutorial explores a unique application of rational exponent rules by focusing on multiplication by 9. You will learn how to re-express 9 as a power of 3, enabling...

TermDefinitionExample Rational ExponentAn exponent that is a fraction, in the form p/q, where p is the power and q is the root. For a base x, x^(p/q) is equivalent to the q-th root of x raised to the power of p, or (q√x)^p.8^(2/3) means the cube root of 8, which is 2, squared. So, 8^(2/3) = (³√8)² = 2² = 4. BaseThe number or variable that is being raised to a power in an exponential expression.In the expression 3^(5/2), the base is 3. Multiplying by 9 as a Power of 3The core technique of this lesson: treating the number 9 not as an integer, but as an exponential term, 3², to facilitate simplification with other base-3 expressions.To simplify 3^(1/2) * 9, we rewrite it as 3^(1/2) * 3². Common BaseWhen two or more exponential terms share the same base number or variable. Having a common bas...

Product of Powers Property x^m * x^n = x^(m+n) When multiplying two exponential expressions with the same base, you keep the base and add the exponents. This is the primary rule used when you multiply by 9 (as 3²). Power of a Power Property (x^m)^n = x^(m*n) When raising an exponential expression to another power, you keep the base and multiply the exponents. This is useful when the term you are multiplying by 9 is itself a power of 3, like (27^x).