Exponents, Roots, and Scientific Notation

Meaning of powers and roots

In the power 4³, 4 is the base, or repeated factor, and 3 is the exponent, or number of factors. Thus 4³=4×4×4=64, not 4×3. A negative base needs grouping: (−3)²=9, but −3² means −(3²)=−9.

A square root asks for the nonnegative number whose square equals the number inside the square-root sign. Therefore √81=9. Although both 9 and −9 solve x²=81, the radical symbol √81 names only the nonnegative root.

Scientific notation and magnitude

Scientific notation has the form a×10ⁿ with 1≤|a|<10. For 4,500,000, move the decimal six places left: 4.5×10⁶. For 0.00072, move it four places right: 7.2×10⁻⁴. The exponent records the movement needed to return to ordinary notation.

To multiply, multiply coefficients and add exponents, then normalize. To divide, divide coefficients and subtract exponents. For addition, first match powers of ten; exponents cannot simply be added.

Check sign and scale

A positive exponent on 10 makes a number at least 10 when the coefficient is at least 1; a negative exponent makes a proper decimal. Squaring a real number cannot give a negative result. Estimate digits: 6.2×10⁵ is about six hundred thousand, not six million.

Worked examples

Worked example 1: (3×10⁴)(2×10³)=6×10⁷. The coefficients give 6, and four plus three gives seven.

Worked example 2: Write 0.000056 in scientific notation. The coefficient is 5.6. Returning to the original requires moving five places left, so 5.6×10⁻⁵.

A reliable way to reason

Count factors for same-base operations: 2³×2⁴=2⁷, while 2⁶÷2²=2⁴. Do not apply these rules to addition: 2³+2⁴=8+16=24, not 2⁷.

For scientific-notation addition, match exponents: 3.2×10⁵+4×10⁴=3.2×10⁵+0.4×10⁵=3.6×10⁵. After multiplication or division, normalize the coefficient: 18×10³=1.8×10⁴. When exponents match, compare coefficients; otherwise check their powers of ten first.

Guided reasoning and error checks

Use expanded meaning to distinguish expressions. The value 3⁴ contains four factors of 3 and equals 81. The value (3²)² contains two copies of 3², also giving four factors of 3 and 81. But 3²+3² is 9+9=18 because addition does not merge repeated factors. For roots, bracket the estimate: since 7²=49 and 8²=64, √55 lies between 7 and 8. This allows a magnitude check even when a root is not a whole number.

In scientific notation, count scale before exact arithmetic. The quotient (6×10⁸)/(2×10³)=3×10⁵. It should be in the hundred-thousands because an eight-power divided by a three-power leaves five powers of ten. Converting 3×10⁵ to 300,000 confirms the scale.

Keep exact values separate from approximations: √55 names an exact value; 7.42 is rounded. Likewise π is exact, while 3.14 is an approximation. Use the precision or approximation the problem requests.

Final self-check

Keep notation unambiguous in written work. Parentheses show whether a negative belongs to the base, and the multiplication sign distinguishes a scientific-notation coefficient from a multi-digit number. These small marks preserve the mathematical claim and make sign or exponent mistakes visible during review.

Your quick summary

• Read powers as repeated multiplication.
• Use perfect-square knowledge to evaluate basic roots.
• Normalize the coefficient and let the exponent record decimal movement.
• Use the exponent sign and an ordinary-number estimate to check the magnitude of scientific notation.