Powers and Roots

Squared means multiply the number by itself.

Cubed means three equal factors.

Which number, multiplied by itself, gives 169?

Find the number that gives 343 when cubed.

The index tells you how many factors of 3 to use.

Multiply four factors of 10.

1. 10^4=10\times10\times10\times10=10000.

The brackets make the whole negative number the base.

1. (-6)^2=(-6)\times(-6)=36. Two negative factors give a positive product.

Multiply each option by itself. Which one gives 0.49?

1. 0.7\times0.7=0.49, so \sqrt{0.49}=0.7.
2. 0.07\times0.07=0.0049, which is too small, and 0.245 is half of 0.49.

Check your answer by multiplying it by itself three times.

1. 0.2\times0.2=0.04, and 0.04\times0.2=0.008. Therefore \sqrt[3]{0.008}=0.2.

Find the square root of the numerator and the square root of the denominator.

1. \sqrt9=3 and \sqrt{25}=5, so \sqrt{\frac{9}{25}}=\frac35.
2. Check: \frac35\times\frac35=\frac{9}{25}.

First find the length of one side.

1. The side length is \sqrt{196}=14 cm.
2. The perimeter is 4\times14=56 cm.

A cube has the same length, width and height.

1. Volume =3.5^3.
2. First 3.5\times3.5=12.25. Then 12.25\times3.5=42.875, so the volume is 42.875 cm³.

A square number can be made by multiplying a whole number by itself.

1. 6^2=36, 7^2=49 and 8^2=64.
2. 24 lies between 4^2 and 5^2; 54 lies between 7^2 and 8^2. Neither is a square number.

Compare 58 with nearby square numbers.

Part (a)

1. 7^2=49 and 8^2=64. Since 49<58<64, the lower whole number is 7.

Part (b)

1. 58<64=8^2, so the upper whole number is 8.

The index tells you how many equal factors to multiply together.

1. 4^3 means three factors of 4: 4\times4\times4=16\times4=64.
2. So Sam is wrong: 4\times3=12 is not the same as 4^3.

List the squares just above 10 squared.

1. 11^2=121, 12^2=144 and 13^2=169. The last is too large; 100 is excluded.
2. The required difference is 144-121=23.

Use the area to find a side before finding the boundary length.

1. One side is \sqrt{225}=15 m. The full perimeter is 4\times15=60 m.
2. Subtract the gate opening: 60-1.5=58.5 m of fence.

Find the side in centimetres first, then convert the length.

1. 40^3=40\times40\times40=64000, so each side is 40 cm.
2. There are 100 cm in a metre, so 40\div100=0.4 m.

List the square numbers in the range, then test their positive roots for primality.

1. 14^2=196 and 18^2=324, so the possible squares are 15^2=225, 16^2=256 and 17^2=289.
2. The numbers 15 and 16 are composite, while 17 is prime. Therefore the required square is 289.

Use the area of one face to find the side length of a cube.

1. Each face is a square, so the side length is \sqrt{36}=6 cm.
2. The volume of one cube is 6^3=6\times6\times6=216 cm³.
3. The cuboid is two cubes, so its volume is 2\times216=432 cm³.