HiSET · Mathematics · 5 min read
Powers, Ratios, Percents, and Unit Conversions
- Apply exponent rules and radical properties to simplify mathematical expressions involving powers, products, quotients, and square roots.
- Convert between fractions, decimals, and percentages, and solve practical problems involving ratios, proportions, unit rates, discounts, sales tax, and percent change.
- Perform accurate unit conversions across both the base-10 metric system and US customary systems of length, weight, and liquid volume.
Understanding Powers, Exponents, and Square Roots
In mathematics, powers provide an efficient way to represent repeated multiplication. For a positive integer exponent, the exponent tells how many times the base appears as a factor. For example, writing a base with an exponent indicates that the base must be multiplied by itself that specific number of times. When dealing with unique exponent values, two fundamental rules apply: any nonzero number raised to the power of zero equals 1, and a nonzero base raised to a negative integer exponent equals the reciprocal of that base raised to the corresponding positive exponent.
Operating with exponents requires following consistent arithmetic properties. When multiplying powers with the same base, keep the base and add the exponents together. Conversely, when dividing powers with the same nonzero base, keep the base and subtract the denominator’s exponent from the numerator’s exponent. Additionally, when raising a power to another power, multiply the exponents together to determine the new exponent on that base.
The sign of the base plays an essential role when raising values to integer powers. A negative base raised to an even integer power yields a positive result, whereas a negative base raised to an odd integer power yields a negative result. This distinction is critical when simplifying expressions with negative signs inside parentheses.
Square roots operate in close connection with exponents. A perfect square is an integer that can be expressed as the square of another integer. The principal square root of a positive real number is its non-negative square root, while negative real numbers do not have real square roots. Squaring a non-negative number and taking its square root are inverse operations. Furthermore, the square root of a product equals the product of the square roots for non-negative values, and the square root of a quotient equals the quotient of the square roots for non-negative values with a non-zero denominator.
Common mistakes
- Dividing by the new amount rather than the positive original amount when calculating a standard percent increase or decrease.
- Multiplying instead of dividing when converting a smaller measurement unit into a larger measurement unit.
- Assuming that a nonzero base raised to a negative exponent yields a negative value, rather than taking the reciprocal of the base raised to the positive exponent.
- Confusing sign rules for exponents by failing to recognize that a negative base raised to an odd integer power yields a negative result, while an even integer power yields a positive result.
Key points
- A positive integer exponent indicates repeated multiplication of the base, any nonzero number raised to the zero power equals 1, and negative exponents indicate reciprocals.
- For non-negative numbers, squaring and taking square roots are inverse operations, with square root properties extending across products and non-zero quotients.
- Percentages represent parts per one hundred; conversions between fractions, decimals, and percentages require dividing, multiplying by powers of ten, or shifting the decimal point two places.
- A proportion equates two ratios, and its unknown values can be solved because the cross products of two equal fractions are equal.
- For a positive original amount, percent change is found by dividing the new amount minus the original amount by the original amount and multiplying by 100.
- Converting from a larger unit to a smaller unit involves multiplication, while converting from a smaller unit to a larger unit involves division across both metric and US customary systems.
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Sample lesson: Rational Numbers and Arithmetic Operations25 practice questions on this exam
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