FN3, FN6, HN3, HN6 - Powers and roots

FN3, FN6, HN3, HN6 - Powers and roots

Read powers as repeated multiplication and use roots to reverse that process. You will recognise useful powers of small numbers, distinguish a square root from the solutions of a square equation, and calculate expressions in the correct order.

Powers

Do 343^4 and 434^3 have the same value? They use the same two digits, but those digits have different jobs.

Three trays each holding three rows of three cakes contain 3×3×3=273\times3\times3=27 cakes. The short way to write that repeated multiplication is 333^3.

Power, base and index

In 343^4, the base is 33 and the index (or exponent) is 44. A positive integer index tells you how many copies of the base to multiply:

34=3×3×3×3.3^4=3\times3\times3\times3.

The second power is called a square; the third power is called a cube.

The names square and cube come from shape. A square of side 33 has area 32=93^2=9 square units; a cube of side 33 has volume 33=273^3=27 cubic units. The index counts factors, not the size of the increase.

Worked example 1: evaluating a power

Question. Work out 343^4.

Plan. Write four factors of 33, then multiply in stages.

34=3×3×3×3four copies of the base=9×9pair the factors=81\begin{aligned} 3^4&=3\times3\times3\times3 &&\text{four copies of the base}\\ &=9\times9 &&\text{pair the factors}\\ &=81 \end{aligned}

Answer. 8181.

Check. 33=273^3=27, and multiplying once more by 33 gives 8181.

Now interchange the base and index. This changes both the repeated factor and how often it appears.

Worked example 2: changing the base and index

Question. Work out 434^3. Is it equal to 343^4?

Plan. Multiply three copies of 44, then compare with 8181.

43=4×4×4three factors, each equal to 4=16×4=64\begin{aligned} 4^3&=4\times4\times4 &&\text{three factors, each equal to }4\\ &=16\times4=64 \end{aligned}

Answer. 43=644^3=64, so 43≠344^3\ne3^4.

Check. 64÷4=1664\div4=16, then 16÷4=416\div4=4: reversing two multiplications leaves the first factor.

Build familiarity with these values by multiplying along each row. Each move right uses the same multiplier again.

BaseFirst powerSquareCubeFourth power
222244881616
33339927278181
444416166464256256
55552525125125625625

For example, 32=2532=2^5 and 243=35243=3^5. A value can be a power in more than one way: 16=24=4216=2^4=4^2. For any positive integer index, a base of 11 gives 11, and a base of 00 gives 00.

Watch out

535^3 is 5×5×5=1255\times5\times5=125, not 5×3=155\times3=15. The index counts repeated factors.

Compare another pair with the roles interchanged.

Roots

A root reverses a power: finding a square root asks which non-negative number squares to give the stated value.

Because 72=497^2=49, we write 49=7\sqrt{49}=7. There is no small index written on a square-root sign. A cube root has a small 33: 1253=5\sqrt[3]{125}=5, because 53=1255^3=125.

Principal root

For a non-negative number, the square-root symbol x\sqrt{\phantom{x}} means the principal square root, which is non-negative. More generally, an even-root symbol means the non-negative real root. An odd real root can be positive, zero or negative: −83=−2\sqrt[3]{-8}=-2.

Squaring either 99 or −9-9 gives 8181. The square-root symbol selects 99, whereas an equation asking for every number whose square is 8181 has two answers.

Worked example 3: a square root and a square equation

Question. (a) Work out 81\sqrt{81}. (b) Find the numbers xx for which x2=81x^2=81.

Plan. Use 92=819^2=81; for the equation, check both signs.

81=9the symbol selects the non-negative root92=81positive solution(−9)2=(−9)×(−9)=81negative solution\begin{aligned} \sqrt{81}&=9 &&\text{the symbol selects the non-negative root}\\ 9^2&=81 &&\text{positive solution}\\ (-9)^2&=(-9)\times(-9)=81 &&\text{negative solution} \end{aligned}

Answer. (a) 99. (b) x=9x=9 or x=−9x=-9.

Check. Both equation answers multiply by themselves to give 8181; only 99 is non-negative.

A cube root reverses three equal factors; a fourth root reverses four. The root's small index tells you which power to check.

Worked example 4: cube and higher roots

Question. Work out (a) −1253\sqrt[3]{-125} and (b) 814\sqrt[4]{81}.

Plan. Find a number with the required cube or fourth power, respecting the root's sign convention.

(−5)3=(−5)×(−5)×(−5)=−12534=3×3×3×3=81\begin{aligned} (-5)^3&=(-5)\times(-5)\times(-5)=-125\\ 3^4&=3\times3\times3\times3=81 \end{aligned}

Answer. (a) −5-5; (b) 33.

Check. A product of three negative factors is negative, as required in (a). Although (−3)4=81(-3)^4=81 too, the fourth-root symbol selects the non-negative value 33.

These are real-number calculations. A negative number has no real square root, because squaring a real number cannot give a negative result. Zero is a boundary case: 0=0\sqrt0=0, and x2=0x^2=0 has just one solution, x=0x=0.

Watch out

Do not write 36=±6\sqrt{36}=\pm6. Write 36=6\sqrt{36}=6. The two values 66 and −6-6 are the solutions of x2=36x^2=36.

Use the inverse power to check each new root.

Operation order

An expression needs a shared order of operations so that everyone calculates the same value.

Work inside brackets first. Then calculate powers and roots. Next do multiplication and division in their written order from left to right. Finally do addition and subtraction from left to right. Multiplication and division have equal priority, as do addition and subtraction.

For example, 24÷6×2=4×2=824\div6\times2=4\times2=8, not 24÷12=224\div12=2. Similarly, 18−7+2=11+2=1318-7+2=11+2=13, not 18−9=918-9=9. A fraction bar also groups its whole numerator and denominator: 8+43=12÷3=4\frac{8+4}{3}=12\div3=4.

Worked example 5: powers and roots before adding

Question. Work out 18−3×4+2518-3\times4+\sqrt{25}.

Plan. Evaluate the root and multiplication before doing the additions and subtractions.

18−3×4+25=18−12+5root and multiplication first=6+5then work left to right=11\begin{aligned} 18-3\times4+\sqrt{25}&=18-12+5 &&\text{root and multiplication first}\\ &=6+5 &&\text{then work left to right}\\ &=11 \end{aligned}

Answer. 1111.

Check. The root adds 55, and the product subtracts 1212, so the net change from 1818 is minus 77, giving 1111.

Now add just one pair of brackets. They make the subtraction happen before the multiplication.

Worked example 6: brackets change the calculation

Question. Work out (18−3)×4+25(18-3)\times4+\sqrt{25}.

Plan. Calculate the bracket and root, multiply, then add.

(18−3)×4+25=15×4+5bracket and root=60+5multiplication=65\begin{aligned} (18-3)\times4+\sqrt{25}&=15\times4+5 &&\text{bracket and root}\\ &=60+5 &&\text{multiplication}\\ &=65 \end{aligned}

Answer. 6565.

Check. Four groups of 18−318-3 give 72−12=6072-12=60; adding 55 gives 6565.

Brackets also decide what a power acts on: (−3)2=9(-3)^2=9 squares the whole negative number, while −32=−9-3^2=-9 means the negative of 323^2. Calculate the power before the minus sign unless brackets include the sign.

Watch out

A memory aid for operation order must not make you do every division before every multiplication. These operations share priority; read them from left to right.

The opening powers also follow a convention: 343^4 means four factors of 33, so it is 8181; 434^3 means three factors of 44, so it is 6464. The same digits need not produce the same value.

Quick Recap

  • A positive integer index counts equal factors; it does not multiply the base by the index.
  • Check a root by raising it to the corresponding power.
  • a\sqrt a is non-negative; solving x2=ax^2=a for positive aa gives both signs.
  • Calculate brackets, then powers and roots, then multiplication/division left to right, then addition/subtraction left to right.

34=813^4=81 and 43=644^3=64: changing the jobs of the digits changes the calculation.