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

1 Product rule ★★★

Simplify each expression.

1. \(x^4\cdot x^7\)
2. \(6a^3\cdot 2a^5\)
3. \((-3m)(5m^4)\)
4. \(y\cdot y^9\)

2 Quotient rule ★★★

Simplify.

1. \(\dfrac{z^{12}}{z^5}\)
2. \(\dfrac{24b^8}{6b^2}\)
3. \(\dfrac{18x^5y^3}{9x^2y}\)

3 Numerical powers ★★★

Compute without a calculator.

1. \(2^3\cdot 2^4\)
2. \((3^2)^2\)
3. \(7^0 + 5^0\)
4. \(\dfrac{10^3}{10^1}\)

4 Power of a power ★★★

Simplify.

1. \((x^3)^5\)
2. \((2y^4)^3\)
3. \((-a^2)^4\)
4. \((xy^2)^3\)

5 Negative exponents as numbers ★★★

Write each value as a fraction or a decimal.

1. \(2^{-3}\)
2. \(10^{-2}\)
3. \(\left(\dfrac13\right)^{-2}\)
4. \((-2)^{-3}\)

6 Scientific notation conversions ★★★

Convert.

1. Write 3,600,000 in scientific notation.
2. Write 0.0072 in scientific notation.
3. Write \(5.1\times 10^4\) in standard form.
4. Write \(9\times 10^{-5}\) in standard form.

7 Name that polynomial ★★★

For each expression, give the number of terms, the name (monomial, binomial, trinomial) and the degree.

1. \(7x\)
2. \(x^2 - 9\)
3. \(4x^3 + x - 1\)
4. \(12\)

8 Adding polynomials ★★★

Add: \((4x^2 - 3x + 8) + (x^2 + 9x - 2)\).

9 Subtracting polynomials ★★★

Subtract: \((7x^2 + 2x - 5) - (3x^2 - 6x + 4)\).

10 Monomial times polynomial ★★★

Expand \(-2x(3x^2 - 5x + 4)\).

11 Product of two binomials ★★★

Expand and simplify.

1. \((x+7)(x+4)\)
2. \((x-6)(x+2)\)
3. \((x-5)(x-3)\)

12 Special products ★★★

Use a special product formula.

1. \((x+9)^2\)
2. \((2x-1)^2\)
3. \((5x+3)(5x-3)\)

13 Only positive exponents ★★★

Simplify and write the answer with positive exponents.

1. \(x^{-4}\cdot x^{9}\)
2. \(6a^{-2}b^3\)
3. \(\dfrac{x^2}{x^8}\)
4. \((3y^{-1})^2\)

14 Scientific notation arithmetic ★★★

Give each result in scientific notation.

1. \((4\times 10^6)(2\times 10^3)\)
2. \(\dfrac{9\times 10^7}{3\times 10^2}\)
3. \((5\times 10^4)(6\times 10^{-2})\)

15 The community garden ★★★

A rectangular garden is \((x+8)\) feet long and \((x+3)\) feet wide.

1. Write its area as a polynomial in standard form.
2. Find the area, in square feet, when \(x = 12\).

16 True or false? ★★★

Decide whether each statement is true or false. Justify, and correct the false ones.

1. \((x+3)^2 = x^2 + 9\)
2. \(x^2\cdot x^3 = x^6\)
3. \((2x)^3 = 2x^3\)
4. \(5^0 = 0\)
5. \((a-b)(a+b) = a^2 - b^2\)

17 A cubic product ★★★

Expand and simplify \((3x-2)(x^2+2x-4)\), then check your result with \(x = 2\).

18 Combined expansion ★★★

Expand and simplify \((x+4)^2 - (x-2)(x+5)\).

19 Mental math with identities ★★★

Use a special product to compute without a calculator.

1. \(62^2\)
2. \(38\times 42\)

20 Simplify a quotient of powers ★★★

Simplify \(\dfrac{(2x^3)^2\cdot 3x^{-1}}{6x^4}\) for \(x \neq 0\).

21 Light from the Sun ★★★

Light travels at about \(3\times 10^8\) meters per second (about \(1.9\times 10^5\) miles per second). The Sun is about \(1.5\times 10^{11}\) meters from Earth.

1. How many seconds does sunlight need to reach Earth? Use scientific notation.
2. Convert to minutes and round to the nearest tenth.

22 A square with a corner removed ★★★

A square sheet of metal has side \((3x+2)\) centimeters. A square corner of side \((x+1)\) centimeters is cut off.

1. Write the remaining area as a polynomial in standard form.
2. Find the area for \(x = 2\).

23 Find the missing numbers ★★★

1. Find numbers \(a\) and \(b\) so that \((x+a)(x+b) = x^2 + 9x + 20\).
2. Find \(k\) and \(m\) so that \((x+k)^2 = x^2 + 14x + m\).