15.7 Power and Root Phrase Translation

Power and Root Phrase Translation catalogs the verbal phrasings that signal exponentiation and root extraction, with particular attention to whether a power or root applies to a single quantity or to an entire grouped sum or difference described in the statement.

Naming Specific Powers

Square of a Quantity

The phrase "the square of a quantity" translates into that quantity raised to the second power, reflecting the standard meaning of squaring as multiplying a number by itself.

Cube of a Quantity

The phrase "the cube of a quantity" translates into that quantity raised to the third power, reflecting the standard meaning of cubing as multiplying a number by itself twice more.

Quantity Raised to a Power

The phrase "a quantity raised to the nth power," for any stated exponent, translates into that quantity with the stated exponent attached, following the general pattern used for square and cube.

Naming Specific Roots

Square Root of a Quantity

The phrase "the square root of a quantity" translates into that quantity placed beneath a radical symbol with no index shown, since an omitted index conventionally means a square root.

Cube Root of a Quantity

The phrase "the cube root of a quantity" translates into that quantity placed beneath a radical symbol with an index of three shown, reflecting the standard notation for a cube root.

Powers and Roots Applied to Grouped Expressions

Power of a Grouped Sum

When a verbal statement describes a power applied to an entire sum, such as "the square of the sum of a number and three," the sum is enclosed in parentheses and the exponent is placed on the entire grouped quantity.

Power of a Grouped Difference

When a verbal statement describes a power applied to an entire difference, such as "the cube of the difference of a number and one," the difference is enclosed in parentheses and the exponent is placed on the entire grouped quantity, exactly as with a grouped sum.

Powers and Roots as Part of a Larger Sum

Sum Containing a Power

When a verbal statement describes a power added to another term, such as "the square of a number, increased by five," the power is applied only to the named quantity, and the resulting term is then added to the rest of the expression, with no grouping needed around the power itself.

Difference Containing a Power

When a verbal statement describes a power subtracted from or subtracting another term, such as "five, decreased by the square of a number," the power again applies only to the named quantity, with the subtraction applied afterward between the resulting term and the rest of the expression.

Roots of Compound Quantities

Root Containing a Compound Radicand

When a verbal statement describes a root applied to an entire sum or difference, such as "the square root of the sum of a number and four," everything under the described sum or difference is placed beneath the radical bar together, since the radical bar itself groups the entire radicand as a single unit.

Confirming Correct Scope

Power Scope Translation

Correctly translating a power phrase requires identifying exactly what the verbal statement intends the exponent to apply to, since a power applied to a single named quantity is written without extra grouping, while a power applied to an entire described sum or difference requires enclosing that sum or difference in parentheses first.

Radical Scope Translation

Correctly translating a root phrase requires identifying exactly what the verbal statement intends to be placed beneath the radical bar, since everything belonging to the described radicand, whether a single quantity or an entire sum or difference, must be captured together underneath that single bar.