Surds

Lesson

It's time to put your calculator abilities to the test and perform some calculations involving surds.

There are many different calculator models throughout the world. Your calculator may have different features or button names to the ones mentioned in this chapter.

Remember that on most scientific or graphics calculators each button has a primary function written on it, and a secondary function written above it. To use the secondary function of any button, you will first have to press $SHIFT$`S``H``I``F``T` (or the equivalent button on your calculator).

By now, you should be very familiar with using the $x^2$`x`2 and $x^3$`x`3 buttons on your calculator to find the square or cube of a number.

We can now use the square root button $\sqrt{\ }$√ and the cube root button $\sqrt[3]{\ }$^{3}√ to calculate simple surds.

For instance, to calculate $\sqrt{7}$√7 we simply press:

$\sqrt{\ }$√ | $7$7 | $=$= |

And our calculator will display:

$2.645751311$2.645751311

Most calculators only display about the first ten digits on the screen, so questions will ask us to round our answer.

To calculate surds involving *any* powers or roots, we use the n^{th }power $x^y$`x``y` and n^{th} root $\sqrt[y]{x}$^{y}√`x` buttons. On some calculators, these buttons will be labelled as $\wedge$∧ and $\sqrt[x]{\ }$^{x}√ .

If we are asked to calculate $17^{0.2}$170.2, we simply press:

$1$1 | $7$7 | $x^y$xy |
$0$0 | $\bullet$• | $2$2 | $=$= |

Or, to calculate $13^{-0.75}$13−0.75, we press:

$1$1 | $3$3 | $x^y$xy |
$-$− | $0$0 | $\bullet$• | $7$7 | $5$5 | $=$= |

And to calculate an n^{th} root such as $\sqrt[5]{29}$^{5}√29, we press:

$5$5 | $\sqrt[y]{x}$^{y}√x |
$29$29 | $=$= |

However, if we use fractions in our calculations, we have to be careful. Brackets will be needed.

Say we're calculating $100^{\frac{1}{2}}$10012. We know that $100^{\frac{1}{2}}=\sqrt{100}$10012=√100$=$=$10$10. However, if we type:

$1$1 | $0$0 | $0$0 | $x^y$xy |
$1$1 | $a\frac{b}{c}$abc |
$2$2 | $=$= |

Most calculators will give the result:

$50$50

This is because it thinks we mean $\frac{100^1}{2}$10012, or "$100$100 to the power of $1$1, divided by $2$2". This is why we need to put brackets to indicate that the entire fraction is a power:

$1$1 | $0$0 | $0$0 | $x^y$xy |
$($( | $1$1 | $a\frac{b}{c}$abc |
$2$2 | $)$) | $=$= |

The same applies to a calculation such as $\sqrt[4]{\frac{3}{8}}$^{4}√38:

$4$4 | $\sqrt[y]{x}$^{y}√x |
$($( | $3$3 | $a\frac{b}{c}$abc |
$8$8 | $)$) | $=$= |

There are two other common situations where we need to remember brackets in our calculations.

The first is whenever we have additions or subtractions under a root sign, such as $\sqrt{\pi+5}$√π+5:

$\sqrt{\ }$√ | $($( | $\pi$π | $+$+ | $5$5 | $)$) | $=$= |

The second is whenever we have additions or subtractions in the numerator or denominator of a fraction, such as $\frac{1}{7+\sqrt{2}}$17+√2:

$1$1 | $a\frac{b}{c}$abc |
$($( | $7$7 | $+$+ | $\sqrt{\ }$√ | $2$2 | $)$) | $=$= |

Find the value of $92^{\frac{1}{3}}$9213 correct to three decimal places.

Find the value of $\sqrt[3]{4.5+8^2}$^{3}√4.5+82 correct to three decimal places.

Find the value of $\frac{\sqrt{2+\pi}}{\sqrt{\pi-1}}$√2+π√π−1 correct to three decimal places.