Indices and standard form worked examples are easier to follow when you separate two questions: what operation is being performed, and how large should the answer be? A negative power changes a number into a reciprocal. It does not automatically make the answer negative. Standard form then uses powers of ten to write a large or small number without a long string of zeros.
This lesson starts with the meaning of a power, builds up to mixed calculations and finishes with practice you can check yourself. All examples are original. Keep exact fractions during the working, and make a rough size check before accepting a calculator display.
Read the base before using an index law
In 73, the base is 7 and the index is 3. The expression means 7 × 7 × 7. In (−7)2, the brackets make −7 the base, so the result is positive 49. Without brackets, −72 means the negative of 7 squared, which is −49.
For the rules below, use a positive base a. This avoids the extra restrictions that arise when fractional powers are applied to negative numbers. A denominator must never be zero.
- Multiplying powers with the same base: am × an = am+n.
- Dividing powers with the same base: am ÷ an = am−n.
- Raising a power to another power: (am)n = amn.
- A zero power: a0 = 1.
- A negative power: a−n = 1/an.
Notice the conditions hidden inside the first two descriptions: the bases must match. You cannot simplify 23 × 54 by adding the indices. Likewise, a2 + a3 is a sum, so the multiplication rule does not apply.
Worked example 1: a negative power inside a quotient
Question: Simplify (34 × 3−2)/33, leaving an exact answer.
- The numerator is a product with base 3, so it becomes 34+(−2) = 32.
- Dividing by 33 gives 32−3 = 3−1.
- The negative index means a reciprocal: 3−1 = 1/3.
Check: The original numerator is 81 × 1/9 = 9. Dividing 9 by 27 gives 1/3 again. The answer is positive and smaller than one. A result of −3 would fail both checks.
When dividing by a negative power, preserve its sign in brackets. For example, 52/5−3 = 52−(−3) = 55. Subtracting a negative index increases the exponent; it is not the same as subtracting 3.
Worked example 2: fractional powers without decimals
A fractional index combines a root and a power. For a positive number a, am/n means take the nth root, then raise the result to power m. Taking the root first usually keeps the arithmetic manageable.
Question: Evaluate 1252/3 × 16−3/4.
- The cube root of 125 is 5. Therefore 1252/3 = 52 = 25.
- The fourth root of 16 is 2. Therefore 163/4 = 23 = 8.
- The negative exponent makes the second factor 1/8.
- The product is 25 × 1/8 = 25/8.
The answer could also be written as 3.125, but the fraction shows that no rounding has occurred. Do not interpret 2/3 as an instruction to multiply 125 by two-thirds. The fraction is an exponent, not a coefficient.
Standard form is a value and a scale
For a positive number, standard form is a × 10n, where 1 ≤ a < 10 and n is an integer. The coefficient a gives the significant digits; the power of ten gives the scale. For negative numbers, the magnitude of the coefficient must lie from 1 up to, but not including, 10.
For example, 6,280,000 = 6.28 × 106, while 0.000628 = 6.28 × 10−4. Check by multiplying back: 10−4 is 0.0001, so 6.28 × 0.0001 is 0.000628.
The expression 62.8 × 105 has the same value as 6.28 × 106, but its coefficient is too large for standard form. Converting the coefficient from 62.8 to 6.28 divides it by ten, so the power-of-ten factor must be multiplied by ten to preserve the value.
Worked example 3: multiply and normalise
Question: Calculate (7.5 × 10−4)(4.8 × 106) in standard form.
First multiply the coefficients: 7.5 × 4.8 = 36. Next multiply the powers of ten: 10−4 × 106 = 102. The intermediate answer is 36 × 102.
Since 36 = 3.6 × 10, the final answer is 3.6 × 103, or 3,600. The normalisation step changes the exponent from 2 to 3. Leaving 36 × 102 would be numerically correct but would not satisfy the requested form.
A rough check uses 0.00075 × 4,800,000. Multiplying a few million by a quantity near one-thousandth should give a few thousand, so 3,600 is reasonable.
Worked example 4: addition needs a common scale
Question: Add 4.6 × 105 and 8.5 × 104.
Rewrite the second number as 0.85 × 105. Now both terms use the same power of ten:
(4.6 + 0.85) × 105 = 5.45 × 105.
The decimal check is 460,000 + 85,000 = 545,000. Adding the exponents would incorrectly treat addition as multiplication. It is the operation between the whole terms, rather than the appearance of powers, that decides which rule you use.
Three practice questions with full solutions
Try each question before reading its solution. For every standard-form answer, check both its numerical value and the range of its coefficient.
Practice 1: mixed indices
Simplify (2−3 × 27)/25.
Solution: Combine the indices to obtain 2−3+7−5 = 2−1 = 1/2. Direct substitution gives (1/8 × 128)/32 = 16/32, confirming the fraction.
Practice 2: divide small numbers
Calculate (9.6 × 10−3)/(3.2 × 10−5).
Solution: Divide coefficients and subtract exponents: (9.6/3.2) × 10−3−(−5) = 3 × 102. The answer is 300. This makes sense because the numerator is hundreds of times larger than the denominator.
Practice 3: a power followed by a conversion
Evaluate 813/4 and write the result in standard form.
Solution: The fourth root of 81 is 3, and 33 = 27. Therefore the answer is 2.7 × 101. The fractional exponent produces an exact integer here.
Questions learners often ask
Why is a negative index not a negative answer?
It tells you to take a reciprocal. With a positive base, both the power and its reciprocal are positive. For instance, 6−2 is 1/36.
Why is a nonzero number to power zero equal to one?
Dividing a power by itself gives one. The quotient law also gives an−n = a0, so these must agree for a nonzero base.
Can I add indices when the bases differ?
No. The same-base multiplication law does not apply. Sometimes you can rewrite both bases as powers of one number, but you must show that conversion first.
What does the denominator of a fractional index tell me?
It specifies the root. In a2/5, the denominator 5 indicates a fifth root and the numerator 2 indicates squaring.
Is 10 × 104 in standard form?
No. The coefficient must be less than 10. Rewrite the value as 1 × 105.
How do I check the sign of a power of ten?
For a positive number below one, its standard-form exponent is negative. For a positive number x with 1 ≤ x < 10, the exponent is zero; values with x ≥ 10 need a positive exponent.
Should I round a standard-form answer immediately?
Only when an accuracy is requested or justified by the data. Keep sufficient digits through intermediate calculations and round once at the end.
Continue your Mathematics revision
Use the Mathematics notes hub to organise your next topic, or visit the Mathematics topical-test collection for further revision resources. Both destinations currently show a login requirement. This lesson’s worked examples and solutions are complete on this page.
WhatsApp Us Now