Indices, roots, standard form and exact calculation Revision Notes

What is Standard Form?

Standard form (or scientific notation) is a way of writing numbers that are too large or too small to be conveniently written in decimal form. It follows the format A × 10^n, where 'A' is a number greater than or equal to 1 and less than 10 (1 ≤ A < 10), and 'n' is an integer (a whole number, positive or negative). The 'n' represents how many places the decimal point has moved. For example, 7,500,000 becomes 7.5 × 10^6, and 0.0000032 becomes 3.2 × 10^-6. The value of A is crucial; it must always be a number with only one non-zero digit before the decimal point.

Converting Numbers to Standard Form

To convert a number to standard form, first, identify the position for the decimal point such that the resulting number (A) is between 1 and 10. For large numbers, move the decimal point to the left until there is only one non-zero digit before it. The number of places moved is 'n', which will be positive. For example, 45,000. Move the decimal 4 places left to get 4.5. So, 45,000 = 4.5 × 10^4. For small numbers (less than 1), move the decimal point to the right until there is one non-zero digit before it. The number of places moved is 'n', which will be negative. For example, 0.00067. Move the decimal 4 places right to get 6.7. So, 0.00067 = 6.7 × 10^-4. Always count the number of places the decimal point moves accurately.

Converting from Standard Form to Ordinary Numbers

To convert a number from standard form (A × 10^n) to an ordinary number, you use the power 'n' to determine how many places and in which direction to move the decimal point in 'A'. If 'n' is positive, move the decimal point 'n' places to the right, adding zeros as necessary. For example, 2.3 × 10^5 means move the decimal 5 places right: 2.30000 -> 230,000. If 'n' is negative, move the decimal point 'n' places to the left, adding zeros as necessary. For example, 7.8 × 10^-3 means move the decimal 3 places left: 0007.8 -> 0.0078. It's helpful to visualise the number of decimal places you are moving rather than just adding zeros. The exponent directs the shift of the decimal point.

Multiplying Numbers in Standard Form

To multiply numbers in standard form, multiply the 'A' parts (the numbers between 1 and 10) together, and add the powers of 10. For example, to calculate $(2 \times 10^3) \times (3 \times 10^4)$: First, multiply the 'A' parts: $2 \times 3 = 6$. Next, add the powers of 10: $10^3 \times 10^4 = 10^{(3+4)} = 10^7$. So, the result is $6 \times 10^7$. Always ensure your final answer is in standard form. If multiplying the 'A' parts results in a number greater than or equal to 10 (e.g., $5 \times 4 = 20$), you'll need to adjust the 'A' part and increase the power of 10 accordingly. For instance, $20 = 2 \times 10^1$, so the exponent would increase by 1.

Dividing Numbers in Standard Form

To divide numbers in standard form, divide the 'A' parts, and subtract the powers of 10. For example, to calculate $(6 \times 10^5) \div (2 \times 10^2)$: First, divide the 'A' parts: $6 \div 2 = 3$. Next, subtract the powers of 10: $10^5 \div 10^2 = 10^{(5-2)} = 10^3$. So, the result is $3 \times 10^3$. Again, check that your final answer is in standard form. If dividing the 'A' parts results in a number less than 1 (e.g., $2 \div 4 = 0.5$), you'll need to adjust the 'A' part and decrease the power of 10. For instance, $0.5 = 5 \times 10^{-1}$, so the exponent would decrease by 1.

Adding and Subtracting Numbers in Standard Form

Adding and subtracting numbers in standard form requires both numbers to have the same power of 10. If they don't, you must rewrite one or both numbers so their power of 10 matches. It's often easiest to convert the smaller power of 10 to the larger power of 10, or to convert both to the smaller power of 10. For example, to calculate $(3 \times 10^4) + (2 \times 10^3)$: Convert $2 \times 10^3$ to $0.2 \times 10^4$. Then, $(3 \times 10^4) + (0.2 \times 10^4) = (3+0.2) \times 10^4 = 3.2 \times 10^4$. Alternatively, convert $3 \times 10^4$ to $30 \times 10^3$. Then, $(30 \times 10^3) + (2 \times 10^3) = (30+2) \times 10^3 = 32 \times 10^3 = 3.2 \times 10^4$ (after adjusting to standard form).

Checking and Adjusting to Standard Form

After any calculation, always verify that your answer is in standard form, i.e., $A \times 10^n$ where $1 \le A < 10$. If the 'A' part is too large (e.g., 25), divide it by 10 and increase the exponent by 1 ($25 \times 10^3 = 2.5 \times 10^1 \times 10^3 = 2.5 \times 10^4$). If the 'A' part is too small (e.g., 0.04), multiply it by 10 and decrease the exponent by 1 ($0.04 \times 10^6 = 4 \times 10^{-2} \times 10^6 = 4 \times 10^4$). This adjustment ensures the number adheres to the definition of standard form.

What is Standard Form?

Standard form is a concise way to write very large or very small numbers. It expresses a number as $A \times 10^n$, where $A$ is a number such that $1 \le A < 10$, and $n$ is an integer (a whole number including negative numbers and zero). For instance, $45000000$ can be written as $4.5 \times 10^7$. Here, $A=4.5$ and $n=7$. For a small number like $0.000082$, it is $8.2 \times 10^{-5}$. Here, $A=8.2$ and $n=-5$. The sign of $n$ indicates whether the number is large (positive $n$) or small (negative $n$). The value of $n$ represents the number of places the decimal point has been shifted.

Multiplying and Dividing Numbers in Standard Form

When multiplying numbers in standard form, multiply the 'A' parts and add the powers of 10. For example, $(2 \times 10^3) \times (4 \times 10^5) = (2 \times 4) \times 10^{(3+5)} = 8 \times 10^8$. When dividing, divide the 'A' parts and subtract the powers of 10. For example, $(8 \times 10^7) \div (2 \times 10^3) = (8 \div 2) \times 10^{(7-3)} = 4 \times 10^4$. After performing the operation, always make sure the result is in correct standard form (i.e., the first part is between 1 and 10), adjusting the power of 10 if necessary. For instance, if you get $24 \times 10^5$, this must be rewritten as $2.4 \times 10^6$ by dividing 24 by 10 and adding 1 to the power of 10.

Adding and Subtracting Numbers in Standard Form

Adding and subtracting numbers in standard form is generally easiest if you first convert them to ordinary numbers (if they are not excessively large or small), perform the addition or subtraction, and then convert the result back to standard form. For example, to calculate $(3 \times 10^4) + (2 \times 10^3)$: convert $3 \times 10^4 = 30000$ and $2 \times 10^3 = 2000$. Then, $30000 + 2000 = 32000$. Convert back to standard form: $3.2 \times 10^4$. Alternatively, you can make the powers of 10 the same before adding/subtracting the 'A' parts. For example, $2 \times 10^3 = 0.2 \times 10^4$. Then $(3 \times 10^4) + (0.2 \times 10^4) = (3+0.2) \times 10^4 = 3.2 \times 10^4$. This method avoids converting to ordinary numbers but requires careful adjustment of one of the 'A' parts and exponents.

Converting from Standard Form and Back

To convert a number like $A \times 10^n$ to ordinary form: if $n$ is positive, move the decimal point in $A$ to the right $n$ places, adding zeros as needed. For example, $5.4 \times 10^3 = 5400$. If $n$ is negative, move the decimal point in $A$ to the left $|n|$ places, adding zeros as needed. For example, $7.1 \times 10^{-2} = 0.071$. To convert an ordinary number to standard form: move the decimal point until there is only one non-zero digit to its left. The number of places moved becomes $n$. If moving the decimal point left to get $A$, $n$ is positive. If moving the decimal point right to get $A$, $n$ is negative. For example, $650000 = 6.5 \times 10^5$. For $0.000302 = 3.02 \times 10^{-4}$.

What are Exact Calculations?

Exact calculations involve operations performed without any approximation or rounding at any stage, ensuring the final result is perfectly precise. This means answers are kept in their original mathematical forms, such as fractions, surds (like √3), or terms involving mathematical constants (like π), rather than being converted into rounded decimals.

Why are Exact Answers Important?

Exact answers maintain full precision, preventing 'rounding errors' that can accumulate throughout a problem. When a problem asks for an 'exact' answer, it specifically means avoiding decimal approximations. Using exact values ensures mathematical integrity and is fundamental for certain areas of mathematics and science.

Common Forms of Exact Answers

Exact answers often appear as simplified fractions (e.g., 1/2, 2/3), surds in simplest form (e.g., √2, 3√5, not √8), or expressions involving π (e.g., 5π, 3π/4). Sometimes, exact answers can be combinations of these, such as 2 + √3 or (1/2)π.

Performing Calculations Exactly

To perform calculations exactly, follow standard rules of arithmetic for fractions, surds, and algebraic expressions. For example, when adding fractions, find a common denominator. When multiplying surds, multiply the numbers inside the root. Only simplify surds or fractions when possible to present the answer in its simplest exact form.