IB · Unit 1 · 1A Exponents
Reduce the coefficients. Write matching variables as repeated factors and cancel. Split a sum or difference over a monomial, then simplify each piece.
Monomial over monomial
Reduce the coefficients. Write the variables as repeated factors. Anything divided by itself is 1.
\(\dfrac{x^5}{x^2} = \dfrac{xxxxx}{xx} = x^3\)
Sum or difference over a monomial
Split the fraction into two pieces, then simplify each piece.
\(\dfrac{A+B}{C} = \dfrac{A}{C} + \dfrac{B}{C}\)
\(\dfrac{6x^4 + 9x^2}{3x} = \dfrac{6x^4}{3x} + \dfrac{9x^2}{3x} = 2x^3 + 3x\)
What students miss
\(\dfrac{6x}{12x^2-18x^4}\)
Factor \(6x\) from both terms in the denominator.
\(\dfrac{6x}{6x(2x-3x^3)}\)
Anything divided by itself is 1.
\(\dfrac{1}{2x-3x^3}\)
Do not cancel a factor from only one term of a sum or difference.
One fraction. Coefficients and exponents only.
123reduce = 4
xxxxxxx
Anything divided by itself is 1.
There are three more x’s in the numerator, so \(x^{3}\).
\(4x^3\)
Answer: \(4x^3\)
155reduce = 3
aaaaaaaa
Anything divided by itself is 1.
There are four more a’s in the numerator, so \(a^{4}\).
\(3a^4\)
Answer: \(3a^4\)
82reduce = 4
yyyyyyyy
Anything divided by itself is 1.
There are six more y’s in the numerator, so \(y^{6}\).
\(4y^6\)
Answer: \(4y^6\)
204reduce = 5
ppppp
Anything divided by itself is 1.
There are three more p’s in the numerator, so \(p^{3}\).
qqqq
Anything divided by itself is 1.
There are two more q’s in the numerator, so \(q^{2}\).
\(5p^3 q^2\)
Answer: \(5p^3 q^2\)
186reduce = 3
xxxxxxxx
Anything divided by itself is 1.
There are two more x’s in the denominator, so \(\dfrac{1}{x^{2}}\).
\(\dfrac{3}{x^2}\)
Answer: \(\dfrac{3}{x^2}\)
93reduce = 3
mmmm
Anything divided by itself is 1.
nnnnn
Anything divided by itself is 1.
There are three more n’s in the numerator, so \(n^{3}\).
\(3n^3\)
Answer: \(3n^3\)
412reduce = \(\dfrac{1}{3}\)
xxxxxxxxxxx
Anything divided by itself is 1.
There are five more x’s in the numerator, so \(x^{5}\).
\(\dfrac{x^5}{3}\)
Answer: \(\dfrac{x^5}{3}\)
46reduce = \(\dfrac{2}{3}\)
yyyyyyy
Anything divided by itself is 1.
There are three more y’s in the numerator, so \(y^{3}\).
\(\dfrac{2y^3}{3}\)
Answer: \(\dfrac{2y^3}{3}\)
Split a sum or difference over a monomial, then simplify each piece.
\(\dfrac{8x^3}{4x} + \dfrac{12x^5}{4x}\)
\(\dfrac{8x^3}{4x} = 2x^2\)
\(\dfrac{12x^5}{4x} = 3x^4\)
\(2x^2 + 3x^4\)
Answer: \(2x^2 + 3x^4\)
\(\dfrac{10y^4}{5y^2} + \dfrac{15y^6}{5y^2}\)
\(\dfrac{10y^4}{5y^2} = 2y^2\)
\(\dfrac{15y^6}{5y^2} = 3y^4\)
\(2y^2 + 3y^4\)
Answer: \(2y^2 + 3y^4\)
\(\dfrac{12a^3}{6a} - \dfrac{18a^5}{6a}\)
\(\dfrac{12a^3}{6a} = 2a^2\)
\(\dfrac{18a^5}{6a} = 3a^4\)
\(2a^2 - 3a^4\)
Answer: \(2a^2 - 3a^4\)
\(\dfrac{9b^2}{3b^2} + \dfrac{6b^5}{3b^2}\)
\(\dfrac{9b^2}{3b^2} = 3\)
\(\dfrac{6b^5}{3b^2} = 2b^3\)
\(3 + 2b^3\)
Answer: \(3 + 2b^3\)
\(\dfrac{16x^5 y}{4x^2 y} - \dfrac{8x^2 y^3}{4x^2 y}\)
\(\dfrac{16x^5 y}{4x^2 y} = 4x^3\)
\(\dfrac{8x^2 y^3}{4x^2 y} = 2y^2\)
\(4x^3 - 2y^2\)
Answer: \(4x^3 − 2y^2\)
\(\dfrac{14p^4 q^2}{7p q} + \dfrac{21p q^3}{7p q}\)
\(\dfrac{14p^4 q^2}{7p q} = 2p^3 q\)
\(\dfrac{21p q^3}{7p q} = 3q^2\)
\(2p^3 q + 3q^2\)
Answer: \(2p^3 q + 3q^2\)
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