Indices, roots, standard form and exact calculation Revision Notes

What are Powers?

A power (or index/exponent) tells you how many times a base number is multiplied by itself. For example, aⁿ means 'a' multiplied by itself 'n' times. For instance, 42 (read as '4 squared') means 4 × 4 = 16. 23 (read as '2 cubed') means 2 × 2 × 2 = 8. It's crucial not to confuse 42 with 4 × 2, which equals 8. The power indicates repeated multiplication, not simple multiplication of base and power.

Calculating Square Roots

The square root of a number is the value that, when multiplied by itself, gives the original number. The symbol used is √. For example, √25 = 5 because 5 × 5 = 25. Every positive number has two square roots, one positive and one negative. However, usually in GCSE mathematics, the term 'square root' refers to the positive square root. For example, √100 = 10, not -10, though (-10)2 = 100 too. You should know common square numbers and their roots, e.g., 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144.

Calculating Cube Roots

The cube root of a number is the value that, when multiplied by itself three times, gives the original number. The symbol for cube root is ∛. For example, ∛27 = 3 because 3 × 3 × 3 = 27. Unlike square roots of positive numbers, a positive number has exactly one real cube root. For example, ∛(-8) = -2 because (-2) × (-2) × (-2) = -8. You should know common cube numbers and their roots, e.g., 1, 8, 27, 64, 125.

Estimation of Powers and Roots

Sometimes, you may need to estimate the value of a power or root, especially for numbers that are not perfect squares or cubes. This involves finding the two perfect squares or cubes that the number lies between. For example, to estimate √50: we know √49 = 7 and √64 = 8. Since 50 is very close to 49, √50 is approximately 7. Also, to estimate the value of (3.1)2, you can round 3.1 to 3, so (3.1)2 is approximately 32 = 9. This skill is useful for checking calculations or when an exact answer isn't required.

Understanding Fractional Indices

A fractional index $a^{m/n}$ means taking the $n$-th root of $a$ and then raising it to the power of $m$. Alternatively, it means raising $a$ to the power of $m$ first and then taking the $n$-th root. Both will yield the same result. The notation is $(\sqrt[n]{a})^m = \sqrt[n]{a^m}$. For example, $64^{2/3}$ can be calculated as $(\sqrt[3]{64})^2 = 4^2 = 16$, or as $\sqrt[3]{64^2} = \sqrt[3]{4096} = 16$. This understanding is vital for simplifying expressions without a calculator.

Understanding Negative Indices

A negative index $a^{-n}$ indicates the reciprocal of $a^n$. So, $a^{-n} = 1/a^n$. For instance, $2^{-3} = 1/2^3 = 1/8$. If you have a fraction raised to a negative power, you can invert the fraction and change the sign of the exponent: $(a/b)^{-n} = (b/a)^n$. For example, $(2/3)^{-2} = (3/2)^2 = 9/4$. This rule is particularly useful for simplifying complex expressions.

Combining Fractional and Negative Indices

When an index is both fractional and negative, for example $a^{-m/n}$, combine both rules. First, deal with the negative sign by taking the reciprocal: $a^{-m/n} = 1/a^{m/n}$. Then, apply the fractional index rule: $1/a^{m/n} = 1/(\sqrt[n]{a})^m$. For example, $125^{-2/3} = 1/125^{2/3} = 1/(\sqrt[3]{125})^2 = 1/5^2 = 1/25$. Mastering this combination is key for higher-level problems.

Applying Index Laws

All standard index laws apply to fractional and negative indices. These include $a^x \times a^y = a^{x+y}$, $a^x \div a^y = a^{x-y}$, and $(a^x)^y = a^{xy}$. These laws allow for the simplification of complex expressions involving powers and roots. For example, $(x^{1/2})^4 = x^{(1/2) \times 4} = x^2$. Another example is $x^{1/3} \times x^{2/3} = x^{(1/3) + (2/3)} = x^1 = x$. Always simplify fully.

Multiplication Law: a^m × a^n = a^(m+n)

When you multiply terms with the same base, you add their indices (or powers). For example, 2³ × 2⁴ = 2^(3+4) = 2⁷. This works because 2³ is 2×2×2 and 2⁴ is 2×2×2×2, so multiplying them gives 2×2×2×2×2×2×2, which is 2⁷. Remember, this law only applies if the bases are identical.

Division Law: a^m ÷ a^n = a^(m-n)

When you divide terms with the same base, you subtract the index of the denominator from the index of the numerator. For example, 5⁶ ÷ 5² = 5^(6-2) = 5⁴. This is because 5⁶⁄5² becomes (5×5×5×5×5×5)⁄(5×5), canceling out two of the fives, leaving 5⁴. Again, this rule requires identical bases.

Power of a Power Law: (a^m)^n = a^(m×n)

When you raise a power to another power, you multiply the indices. For example, (3²)³ = 3^(2×3) = 3⁶. This means (3²)³ is (3×3) × (3×3) × (3×3), which is 3 multiplied by itself 6 times. This law is very useful for simplifying expressions involving nested powers.

Zero Index Law: a⁰ = 1 (for a ≠ 0)

Any non-zero number raised to the power of zero is equal to 1. For example, 7⁰ = 1, (x)⁰ = 1. This can be understood using the division law: a^m ÷ a^m = a^(m-m) = a⁰. Since a^m ÷ a^m is also 1 (as long as a^m is not zero), it follows that a⁰ = 1. Note that 0⁰ is undefined.

Multiplying and Dividing Powers with the Same Base

When you multiply terms with the same base, you add their indices. For example, a³ × a⁵ = a³⁺⁵ = a⁸. Think of it as having three 'a's multiplied together, and then another five 'a's multiplied together, resulting in eight 'a's in total. When you divide terms with the same base, you subtract their indices. For example, b⁷ ÷ b² = b⁷⁻² = b⁵. This is like cancelling out common factors; if you have seven 'b's on top and two 'b's on the bottom, two 'b's cancel out, leaving five 'b's on top.

Power of a Power Rule

When an expression with an index is raised to another power, you multiply the indices. For example, (c⁴)³ = c⁴ˣ³ = c¹². This means c⁴ is multiplied by itself three times: c⁴ × c⁴ × c⁴. Using the multiplication rule, this becomes c⁴⁺⁴⁺⁴ = c¹². This rule significantly simplifies such expressions. Remember to apply the outer power to both the coefficient and the variable if there is one, e.g., (2x³)². In this case, it becomes 2² * (x³)², which simplifies to 4x⁶.

Zero and Negative Indices

Any non-zero number or variable raised to the power of zero is always 1. For example, 5⁰ = 1 and y⁰ = 1 (provided y ≠ 0). This holds true because, according to the division rule, aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰, and anything divided by itself is 1. A negative index indicates a reciprocal. For example, x⁻² = 1/x². This means 1 divided by x raised to the positive power. If you have 1/x⁻³, it becomes x³. These rules are crucial for simplifying expressions and understanding the full range of index operations.