### All HSPT Math Resources

## Example Questions

### Example Question #561 : Problem Solving

You are given that are whole numbers.

Which of the following is true of if and are both odd?

**Possible Answers:**

is always odd.

is always odd if is even, and always even if is odd.

is always even.

None of the other statements are true.

is always odd if is odd, and always even if is even.

**Correct answer:**

is always odd if is even, and always even if is odd.

If is odd, then is odd, since the product of two odd whole numbers must be odd. When the odd number is added, the result, , is even, since the sum of two odd numbers must be even.

If is even, then is even, since the product of an odd number and an even number must be even. When the odd number is added, the result, , is odd, since the sum of an odd number and an even number must be odd.

### Example Question #1 : How To Simplify Expressions

Simplify the expression:

**Possible Answers:**

**Correct answer:**

Combine all the like terms.

The terms can be combined together, which gives you .

When you combine the terms together, you get .

There is only one term so it doesn't get combined with anything. Put them all together and you get

.

### Example Question #2 : How To Simplify Expressions

Simplify the following expression:

**Possible Answers:**

**Correct answer:**

First distribute the 2:

Combine the like terms:

### Example Question #1 : How To Simplify Expressions

Simplify the expression:

**Possible Answers:**

*x*

*x* + 1

2*x* + 1

2*x*

*x*^{2 }+ 2*x* + 1

**Correct answer:**

*x* + 1

Factor out a (2*x*) from the denominator, which cancels with (2*x*) from the numerator. Then factor the numerator, which becomes (*x *+ 1)(*x *+ 1), of which one of them cancels and you're left with (*x *+ 1).

### Example Question #21 : Simplifying Expressions

Simplify the following expression: x^{3} - 4(x^{2} + 3) + 15

**Possible Answers:**

**Correct answer:**

To simplify this expression, you must combine like terms. You should first use the distributive property and multiply -4 by x^{2 }and -4 by 3.

x^{3} - 4x^{2} -12 + 15

You can then add -12 and 15, which equals 3.

You now have x^{3} - 4x^{2} + 3 and are finished. Just a reminder that x^{3} and 4x^{2 }are not like terms as the x’s have different exponents.

### Example Question #2 : How To Simplify Expressions

Simplify the following expression:

2x(x^{2} + 4ax – 3a^{2}) – 4a^{2}(4x + 3a)

**Possible Answers:**

12a^{3 }– 16a^{2}x + 8ax^{2} + 2x^{3}

–12a^{3 }– 14ax^{2} + 2x^{3}

–12a^{3 –} 14a^{2}x + 2x^{3}

–12a^{3 }– 22a^{2}x + 8ax^{2} + 2x^{3}

12a^{3 }– 22a^{2}x + 8ax^{2} + 2x^{3}

**Correct answer:**

–12a^{3 }– 22a^{2}x + 8ax^{2} + 2x^{3}

Begin by distributing each part:

2x(x^{2} + 4ax – 3a^{2}) = 2x * x^{2} + 2x * 4ax – 2x * 3a^{2} = 2x^{3} + 8ax^{2} – 6a^{2}x

The second:

–4a^{2}(4x + 3a) = –16a^{2}x – 12a^{3}

Now, combine these:

2x^{3} + 8ax^{2} – 6a^{2}x – 16a^{2}x – 12a^{3}

The only common terms are those with a^{2}x; therefore, this reduces to

2x^{3} + 8ax^{2} – 22a^{2}x – 12a^{3}

This is the same as the given answer:

–12a^{3 }– 22a^{2}x + 8ax^{2} + 2x^{3}

### Example Question #23 : Simplifying Expressions

Which of the following does not simplify to ?

**Possible Answers:**

All of these simplify to

**Correct answer:**

5x – (6x – 2x) = 5x – (4x) = x

(x – 1)(x + 2) - x^{2} + 2 = x^{2} + x – 2 – x^{2} + 2 = x

x(4x)/(4x) = x

(3 – 3)x = 0x = 0

### Example Question #1 : How To Simplify Expressions

Simplify:

**Possible Answers:**

**Correct answer:**

In order to simplify this expression, distribute and multiply the outer term with the two inner terms.

### Example Question #3 : How To Simplify Expressions

Simplify:

**Possible Answers:**

**Correct answer:**

When the same bases are multiplied, their exponents can be added. Similarly, when the bases are divided, their exponents can be subtracted. Apply this rule for the given problem.

### Example Question #4 : How To Simplify Expressions

Simplify:

**Possible Answers:**

**Correct answer:**

To simplify this expression, reduce the term inside the parenthesis.

Rewrite the negative exponent as a fraction.

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