Q11. Write each number in scientific notation.

Background

Topic: Scientific Notation

This question tests your ability to convert large and small numbers into scientific notation, which is a way of expressing numbers as a product of a coefficient and a power of ten.

Key Terms and Formulas:

• Scientific notation: , where and is an integer.

• To convert a number, move the decimal point so that only one nonzero digit remains to the left of the decimal.

• Count how many places you move the decimal to determine the exponent .

Step-by-Step Guidance

1. For 340,200,000: Move the decimal point to the left until only one digit remains to the left. Count the number of places moved.

2. Express the number as using the coefficient and exponent you found.

3. For 0.00000059: Move the decimal point to the right until only one digit remains to the left. Count the number of places moved.

4. Express the number as using the coefficient and exponent you found.

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Q12. Write each number in standard form.

Background

Topic: Scientific Notation to Standard Form

This question tests your ability to convert numbers from scientific notation back to their standard decimal form.

Key Terms and Formulas:

• Standard form: The usual way of writing numbers (not using exponents).

• To convert to standard form, move the decimal point places to the right (if is positive) or left (if $n$ is negative).

Step-by-Step Guidance

1. For : Move the decimal point 3 places to the left.

2. Write out the number with the new decimal placement.

3. For : Move the decimal point 6 places to the right.

4. Write out the number with the new decimal placement.

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Q13. Simplify the following expressions involving exponents.

Background

Topic: Laws of Exponents

This question tests your understanding of exponent rules, including multiplying, dividing, and simplifying expressions with positive and negative exponents.

Key Terms and Formulas:

• Product of powers:

• Quotient of powers:

• Power of a power:

• Negative exponent:

Step-by-Step Guidance

1. Identify the base and the exponents in each term.

2. Apply the appropriate exponent rule (product, quotient, power of a power, negative exponent).

3. Combine like terms and simplify the expression as much as possible.

4. If there are negative exponents, rewrite them as reciprocals.

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Q14. Find each product (multiply polynomials).

Background

Topic: Multiplying Polynomials

This question tests your ability to multiply polynomials using the distributive property and the FOIL method (for binomials).

Key Terms and Formulas:

• Distributive property:

• FOIL method: For , multiply First, Outside, Inside, Last terms.

• Combine like terms after multiplying.

Step-by-Step Guidance

1. Write out each polynomial and identify the terms.

2. Multiply each term in the first polynomial by each term in the second polynomial.

3. Use the distributive property or FOIL method as appropriate.

4. Combine like terms to simplify the expression.

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Q20. Divide each of the following (polynomial division).

Background

Topic: Polynomial Division

This question tests your ability to divide polynomials, which may involve long division or synthetic division, and simplifying the result.

Key Terms and Formulas:

• Division of polynomials:

• Arrange terms in descending order of degree.

• Subtract multiples of the divisor from the dividend to find the quotient and remainder.

Step-by-Step Guidance

1. Write the dividend and divisor in standard form (descending powers).

2. Divide the leading term of the dividend by the leading term of the divisor to get the first term of the quotient.

3. Multiply the divisor by this term and subtract from the dividend.

4. Repeat the process with the new dividend until the degree of the remainder is less than the degree of the divisor.

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