Exponent Rules and Polynomials: practice solutions, Grade 9 – download the PDF

1 Product rule ★★★

Add the exponents of the same base and multiply the coefficients.

1. \(x^{4+7} = x^{11}\).
2. \(6\cdot 2 = 12\) and \(a^{3+5} = a^8\), so \(12a^8\).
3. \((-3)(5) = -15\) and \(m^{1+4} = m^5\), so \(-15m^5\).
4. \(y = y^1\), so \(y^{1+9} = y^{10}\).

2 Quotient rule ★★★

1. \(z^{12-5} = z^7\).
2. \(24\div 6 = 4\) and \(b^{8-2} = b^6\), so \(4b^6\).
3. \(18\div 9 = 2\), \(x^{5-2} = x^3\), \(y^{3-1} = y^2\), so \(2x^3y^2\).

3 Numerical powers ★★★

1. \(2^{3+4} = 2^7 = 128\).
2. \((3^2)^2 = 3^4 = 81\).
3. Each zero power is 1, so \(1 + 1 = 2\).
4. \(10^{3-1} = 10^2 = 100\).

4 Power of a power ★★★

1. \(x^{3\cdot 5} = x^{15}\).
2. \(2^3\cdot y^{12} = 8y^{12}\).
3. \((-1)^4 a^{8} = a^8\) (an even power of a negative is positive).
4. \(x^3 y^{6}\).

5 Negative exponents as numbers ★★★

1. \(2^{-3} = \dfrac{1}{2^3} = \dfrac18\).
2. \(10^{-2} = \dfrac{1}{100} = 0.01\).
3. \(\left(\dfrac13\right)^{-2} = \dfrac{1}{(1/3)^2} = \dfrac{1}{1/9} = 9\).
4. \((-2)^{-3} = \dfrac{1}{(-2)^3} = -\dfrac18\).

6 Scientific notation conversions ★★★

1. \(3.6\times 10^{6}\) (6 places).
2. \(7.2\times 10^{-3}\) (3 places to the right).
3. \(5.1\times 10^4 = 51{,}000\).
4. \(9\times 10^{-5} = 0.00009\).

7 Name that polynomial ★★★

1. One term: monomial, degree 1.
2. Two terms: binomial, degree 2.
3. Three terms: trinomial, degree 3.
4. One term: monomial, degree 0 (a constant).

8 Adding polynomials ★★★

Group like terms: \((4x^2 + x^2) + (-3x + 9x) + (8 - 2)\).

Result: \(5x^2 + 6x + 6\).

9 Subtracting polynomials ★★★

Change every sign of the second polynomial: \(7x^2 + 2x - 5 - 3x^2 + 6x - 4\).

Group: \(4x^2 + 8x - 9\).

10 Monomial times polynomial ★★★

Distribute \(-2x\): \(-2x\cdot 3x^2 = -6x^3\), \(-2x\cdot(-5x) = +10x^2\), \(-2x\cdot 4 = -8x\).

Result: \(-6x^3 + 10x^2 - 8x\).

11 Product of two binomials ★★★

1. \(x^2 + 4x + 7x + 28 = x^2 + 11x + 28\).
2. \(x^2 + 2x - 6x - 12 = x^2 - 4x - 12\).
3. \(x^2 - 3x - 5x + 15 = x^2 - 8x + 15\).

12 Special products ★★★

1. \(x^2 + 2\cdot 9\cdot x + 81 = x^2 + 18x + 81\).
2. \((2x)^2 - 2\cdot 2x\cdot 1 + 1 = 4x^2 - 4x + 1\).
3. \((5x)^2 - 3^2 = 25x^2 - 9\).

13 Only positive exponents ★★★

1. \(x^{-4+9} = x^5\).
2. Only \(a\) has a negative exponent: \(\dfrac{6b^3}{a^2}\).
3. \(x^{2-8} = x^{-6} = \dfrac{1}{x^6}\).
4. \(3^2\cdot y^{-2} = \dfrac{9}{y^2}\).

14 Scientific notation arithmetic ★★★

1. \(8\times 10^{6+3} = 8\times 10^9\).
2. \(3\times 10^{7-2} = 3\times 10^5\).
3. \(30\times 10^{2} = 3\times 10^{1}\times 10^2 = 3\times 10^3\) (the 30 had to be rewritten as \(3\times 10\)).

15 The community garden ★★★

1. Area = length \(\times\) width \(= (x+8)(x+3) = x^2 + 3x + 8x + 24 = x^2 + 11x + 24\).
2. With \(x = 12\): \(144 + 132 + 24 = 300\). Check: the garden is 20 ft by 15 ft, and \(20\times 15 = 300\). The area is 300 square feet (about 27.9 square meters).

16 True or false? ★★★

1. False: the middle term is missing. \((x+3)^2 = x^2 + 6x + 9\); test \(x = 1\): \(16\) versus \(10\).
2. False: exponents are added, so \(x^2\cdot x^3 = x^5\).
3. False: the 2 is cubed too, so \((2x)^3 = 8x^3\).
4. False: any nonzero number to the power 0 equals 1, so \(5^0 = 1\).
5. True: the middle terms \(+ab\) and \(-ab\) cancel.

17 A cubic product ★★★

Distribute: \(3x\cdot x^2 = 3x^3\), \(3x\cdot 2x = 6x^2\), \(3x\cdot(-4) = -12x\), \(-2\cdot x^2 = -2x^2\), \(-2\cdot 2x = -4x\), \(-2\cdot(-4) = 8\).

Combine: \(3x^3 + 4x^2 - 16x + 8\).

Check at \(x = 2\): factored form gives \((4)(4+4-4) = 16\); expanded form gives \(24 + 16 - 32 + 8 = 16\). The results match.

18 Combined expansion ★★★

\((x+4)^2 = x^2 + 8x + 16\) and \((x-2)(x+5) = x^2 + 3x - 10\).

Subtract, flipping every sign of the second result: \(x^2 + 8x + 16 - x^2 - 3x + 10 = 5x + 26\).

19 Mental math with identities ★★★

1. \(62^2 = (60+2)^2 = 3600 + 240 + 4 = 3844\).
2. \(38\times 42 = (40-2)(40+2) = 1600 - 4 = 1596\).

20 Simplify a quotient of powers ★★★

Numerator: \((2x^3)^2 = 4x^6\), then \(4x^6\cdot 3x^{-1} = 12x^{5}\).

Divide: \(\dfrac{12x^5}{6x^4} = 2x^{5-4} = 2x\).

21 Light from the Sun ★★★

1. Time = distance \(\div\) speed \(= \dfrac{1.5\times 10^{11}}{3\times 10^{8}} = 0.5\times 10^{3} = 5\times 10^{2}\) seconds.
2. \(500 \div 60 \approx 8.3\). Sunlight takes about 8.3 minutes.

22 A square with a corner removed ★★★

1. \((3x+2)^2 = 9x^2 + 12x + 4\) and \((x+1)^2 = x^2 + 2x + 1\). Remaining area: \(9x^2 + 12x + 4 - x^2 - 2x - 1 = 8x^2 + 10x + 3\).
2. For \(x = 2\): \(8\cdot 4 + 20 + 3 = 55\). Check: \(8^2 - 3^2 = 64 - 9 = 55\). The area is 55 square centimeters.

23 Find the missing numbers ★★★

1. We need \(a+b = 9\) and \(ab = 20\). The pair 4 and 5 works: \(4+5 = 9\), \(4\cdot 5 = 20\). So \((x+4)(x+5)\).
2. The middle term is \(2k = 14\), so \(k = 7\); then \(m = k^2 = 49\).