Multiply by 10

Learning Objectives Apply the product rule of exponents to expressions with base 10 and rational exponents. Convert expressions between radical form (e.g., √10) and rational exponent form (10^(1/2)). Simplify expressions involving multiplication by 10 and its rational powers. Connect the concept of multiplying by 10 to transformations of the common logarithm function. Solve equations where the variable is part of a rational exponent on a base of 10. Analyze the effect of multiplying by 10^n on a number in scientific notation, where n is a rational number. Ever wonder why an earthquake of magnitude 7 is 10 times more powerful than a 6, not just a little bit stronger? 🤔 It's all about the power of multiplying by 10! This tutorial moves beyond simple arithmetic to explor...

TermDefinitionExample Rational ExponentAn exponent expressed as a fraction m/n, where the numerator 'm' represents a power and the denominator 'n' represents a root. The general form is a^(m/n) = ⁿ√(aᵐ).10^(2/3) is equivalent to the cube root of 10 squared, or ³√(10²). Base 10In the expression 10^x, the number 10 is the 'base'. Our number system is base-10, making it fundamental in scientific notation and common logarithms.In 10^(1/2), the base is 10. Product Rule of ExponentsWhen multiplying two exponential expressions that share the same base, you keep the base and add the exponents.10^a * 10^b = 10^(a+b). For instance, 10^(1/2) * 10^(1/3) = 10^(1/2 + 1/3) = 10^(5/6). Common LogarithmA logarithm with base 10, written as log(x) or log₁₀(x). It answers the qu...

Product Rule for Base 10 10^a ⋅ 10^b = 10^(a+b) To multiply two powers of 10, keep the base 10 and add the exponents. This rule is the foundation of this lesson and applies to all rational exponents 'a' and 'b'. Rational Exponent Definition 10^(m/n) = (ⁿ√10)ᵐ = ⁿ√(10ᵐ) This rule defines how to interpret a fractional exponent. The denominator 'n' becomes the index of the root, and the numerator 'm' becomes the power. Logarithm of a Product by 10 log(10 ⋅ x) = log(10) + log(x) = 1 + log(x) Multiplying a value by 10 inside a common logarithm is equivalent to adding 1 to the logarithm of the original value. This shows a direct link between multiplication and addition via logarithms.