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Questions and Answers
Which expression is equivalent to $7 \times 7 \times 7 \times 7 \times 7$ using index notation?
Which expression is equivalent to $7 \times 7 \times 7 \times 7 \times 7$ using index notation?
- $7^5$ (correct)
- $7^6$
- $5^7$
- $7^4$
What is the value of $3^4$?
What is the value of $3^4$?
- 81 (correct)
- 64
- 7
- 12
Which of the following expressions represents the prime factorisation of 36, expressed in index form?
Which of the following expressions represents the prime factorisation of 36, expressed in index form?
- $2 \times 3 \times 6$
- $2^2 \times 3^2$ (correct)
- $2^2 \times 9$
- $4 \times 3^2$
What is the base in the expression $4^7$?
What is the base in the expression $4^7$?
Simplify the expression $x^5 \times x^3$ using index laws.
Simplify the expression $x^5 \times x^3$ using index laws.
Simplify the expression $a^{12} \div a^4$ using index laws.
Simplify the expression $a^{12} \div a^4$ using index laws.
Which of the following is equivalent to $9^5 \times 9^2$?
Which of the following is equivalent to $9^5 \times 9^2$?
Which expression is equivalent to $z^{15} \div z^5$?
Which expression is equivalent to $z^{15} \div z^5$?
Simplify $(b^3)^4$ using index laws.
Simplify $(b^3)^4$ using index laws.
What is the value of $x^0$, where $x \neq 0$?
What is the value of $x^0$, where $x \neq 0$?
Which expression is equivalent to $(5^2)^3$?
Which expression is equivalent to $(5^2)^3$?
What is the simplified form of $(mn)^0$, assuming $m \neq 0$ and $n \neq 0$?
What is the simplified form of $(mn)^0$, assuming $m \neq 0$ and $n \neq 0$?
Simplify the expression $(3y)^4$?
Simplify the expression $(3y)^4$?
Which expression is equivalent to $\left(\frac{x}{5}\right)^3$?
Which expression is equivalent to $\left(\frac{x}{5}\right)^3$?
Simplify $(2x^2)^3$.
Simplify $(2x^2)^3$.
Which of the following is equivalent to $\left(\frac{a}{4}\right)^2$?
Which of the following is equivalent to $\left(\frac{a}{4}\right)^2$?
Express $4^{-3}$ using a positive index.
Express $4^{-3}$ using a positive index.
Which of the following is equivalent to $\frac{1}{7^{-2}}$?
Which of the following is equivalent to $\frac{1}{7^{-2}}$?
Evaluate $5^{-2}$.
Evaluate $5^{-2}$.
Which of the following is equal to $x^{-5}$?
Which of the following is equal to $x^{-5}$?
Express 0.000075 in scientific notation.
Express 0.000075 in scientific notation.
Which number is equivalent to $3.8 \times 10^6$?
Which number is equivalent to $3.8 \times 10^6$?
What is $9 \times 10^{-4}$ expressed as a basic numeral?
What is $9 \times 10^{-4}$ expressed as a basic numeral?
Which of the following numbers is written in scientific notation?
Which of the following numbers is written in scientific notation?
How many significant figures are there in the number 0.004020?
How many significant figures are there in the number 0.004020?
Round the number 45,678 to three significant figures.
Round the number 45,678 to three significant figures.
Express 0.0003082 in scientific notation to three significant figures.
Express 0.0003082 in scientific notation to three significant figures.
How many significant figures are in $2.050 \times 10^3$?
How many significant figures are in $2.050 \times 10^3$?
Simplify the expression $y^8 \times y^{-2}$ using index laws.
Simplify the expression $y^8 \times y^{-2}$ using index laws.
Which expression is equivalent to $b^{-4} \div b^{-1}$?
Which expression is equivalent to $b^{-4} \div b^{-1}$?
Simplify the expression $(a^{-2})^3$ using index laws.
Simplify the expression $(a^{-2})^3$ using index laws.
Which of the following is equivalent to $25^{-0.5}$?
Which of the following is equivalent to $25^{-0.5}$?
Express the number 52800000000 in scientific notation.
Express the number 52800000000 in scientific notation.
What is 0.0000000361 in scientific notation?
What is 0.0000000361 in scientific notation?
Round 187,529 to two significant figures.
Round 187,529 to two significant figures.
Express 49,632 in scientific notation to three significant figures.
Express 49,632 in scientific notation to three significant figures.
Simplify the expression $\frac{6x^5y^3}{2x^2y}$ using index laws.
Simplify the expression $\frac{6x^5y^3}{2x^2y}$ using index laws.
Which expression is equivalent to $ (4a^2b)^3 $?
Which expression is equivalent to $ (4a^2b)^3 $?
Flashcards
Index Notation
Index Notation
A way of writing repeated multiplication using a base and an index (or power/exponent).
Base (in index notation)
Base (in index notation)
The factor that is repeatedly multiplied in index notation.
Index (or Power/Exponent)
Index (or Power/Exponent)
The number indicating how many times the base is multiplied by itself.
Prime Factorisation
Prime Factorisation
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Prime Number
Prime Number
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Index Form
Index Form
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Expanded Form
Expanded Form
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Index Law for Multiplication
Index Law for Multiplication
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Index Law for Division
Index Law for Division
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Index Law for Power of a Power
Index Law for Power of a Power
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The Zero Index
The Zero Index
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Power of a Product
Power of a Product
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Power of a Fraction
Power of a Fraction
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Negative Index
Negative Index
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Scientific Notation
Scientific Notation
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Significant Figures
Significant Figures
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Study Notes
Index Notation
- Index notation represents repeated multiplication.
- It allows conversion between expanded form and index form.
- It allows evaluation of expressions written in index form.
- Numbers can be expressed as a product of prime factors.
- The expression 5 × 5 × 5, which equals 125, can be written as 53 using index notation.
- In 53, 5 is the base, and 3 is the index (also called power or exponent).
- 125 is a power of 5.
- The powers of 5 are 5, 25, 125, 625, and so on.
- Expanded form: shows the repeated multiplication (e.g., 2 × 2 × 2 × 2 × 2).
- Index form: the base and index (e.g., 25).
- Base: the factor in the product.
- Index: indicates how many times the base appears in the product.
- 22 is read as '2 to the power of 2' or '2 squared,' which equals 4.
- 23 is read as '2 to the power of 3' or '2 cubed,' which equals 8.
- a1 = a, for example: 51 = 5.
- 32 does not mean 3 × 2 = 6.
- Prime factorization involves writing a number as a product of its prime factors.
- A prime number has only two factors: 1 and itself.
Index Laws for Multiplying and Dividing
- Index laws simplify expressions when multiplying or dividing numbers with the same base.
- When multiplying numbers with the same base, add the indices: am × an = am+n.
- When dividing numbers with the same base, subtract the indices: am ÷ an = am−n.
The Zero Index and Power of a Power
- The index law for raising a power to another power is (am)n = am×n = amn
- Any term (except 0) raised to the power of zero is 1: a0 = 1, where a ≠ 0.
- When raising a term in index form to another power, keep the base and multiply the indices.
- A power outside brackets applies only to the expression inside the brackets.
Index Laws Extended
- Power of a product: (a × b)m = (ab)m = ambm, where each number in the brackets is raised to the power of m.
- Power of a fraction: (a/b)m = am/bm, where b ≠ 0, and each number in the brackets is raised to the power of m.
Negative Indices
- Negative indices can be expressed using positive indices: a−m = 1/am and am = 1/a−m (where a ≠ 0).
- a raised to the power −m is the reciprocal of a raised to the power m.
- All index laws apply to expressions with negative indices.
Scientific Notation
- Scientific notation represents very large and very small numbers using powers of 10.
- Numbers in scientific notation are expressed as a × 10m, where a is between 1 and 10 (or greater than −10 to −1), and m is an integer.
- Large numbers are written in scientific notation using positive powers of 10.
- Small numbers are written in scientific notation using negative powers of 10.
Scientific Notation Using Significant Figures
- Significant figures indicate the accuracy of a number.
- Count significant figures from left to right, starting at the first non-zero digit.
- Zeros at the end of a number are counted for decimals but not necessarily for whole numbers.
- In scientific notation, the first significant figure is to the left of the decimal point.
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