Algebra

NZ Level 6 (NZC) Level 1 (NCEA)

Simplify algebraic fractions

Lesson

Simplifying algebraic fractions usually involves the cancelling of common factors between the numerator and denominator. This could include both numeric and algebraic factors.

$\frac{3xy}{6xz}$3xy6xz |
$=$= | $\frac{xy}{2xz}$xy2xz Cancel the highest common numerical factor |

$=$= | $\frac{y}{2z}$y2z Cancel the highest common powers of each pronumeral |

When trying to simplify expressions that involve algebraic fractions, it is best to initially cancel out any immediately recognisable common factors between the numerator and denominator. This should make the simplification process of a bigger expression a lot easier, and we can see this in the example below.

$\frac{x^2}{x}+3x$x2x+3x |
$=$= | $\frac{x^2}{x}+\frac{3x^2}{x}$x2x+3x2x |
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$\frac{x^2}{x}+3x$x2x+3x |
$=$= | $x+3x$x+3x |
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$=$= | $\frac{x^2+3x^2}{x}$x2+3x2x |
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$=$= | $4x$4x |
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$=$= | $\frac{4x^2}{x}$4x2x |
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$=$= | $4x$4x |
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Express $\frac{72s^2t^2}{81s^3t}$72`s`2`t`281`s`3`t` in simplest form.

Yuri wanted to know whether he is faster than his brother Jimmy at building a wooden shelf. Their woodworking instructor told Yuri that his speed was given by the expression $18xy$18`x``y` and that Jimmy's speed was given by $6xy$6`x``y`, where $x$`x` relates to the quality of the tools they use, and $y$`y` depends on the quality of the materials used.

If both brothers use the same tools and materials, write a fraction which represents the ratio of Yuri and Jimmy's construction speeds respectively. Do not simplify your answer.

$\frac{\editable{}}{\editable{}}$

By simplifying the fraction from the previous part, determine how many times faster Yuri is than Jimmy.

If both brothers always use the same tools and materials, does the quality of either one impact how many times faster Yuri is than Jimmy?

No

AYes

BNo

AYes

B

Simplify the following expression:

$\frac{12y^3}{6y^2}+\frac{27y^2}{9y}+\frac{4y}{4}$12`y`36`y`2+27`y`29`y`+4`y`4

Form and solve linear equations and inequations, quadratic and simple exponential equations, and simultaneous equations with two unknowns

Generalise the properties of operations with rational numbers, including the properties of exponents

Apply algebraic procedures in solving problems