Operations with Science

Scientific Notation & Operations With Exponents

Scientific Notation

• Scientific notation (SN) is a compact way of representing really small or really large measurements. The compactness allows us to perform operations on these measurements with ease.

• Measurements written in SN have the same value as the standard form. For example, 500 in SN equals 5.0 x 10⁵.

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Measurements Greater Than One in Scientific Notation

• C: Place a decimal between the first non-zero digit and the remaining digits.

• n: Count the number of digits after the first non-zero - including the zeros.

• Multiply C by 10 to the n power. n is positive since the measurement is greater than 1.

Measurements Less Than One in Scientific Notation

• C: Place a decimal between the first non-zero digit and the remaining digits.

• n: Count the number of zeros that come before the first non-zero digit. Add one to this.

• Multiply C by 10 to the n power. n will be negative since the measurement is less than 1.

Multiple Choice

What is 154000 in scientific notation?

• 154 x 10³

• 15.4 x 10⁴

• 1.54 x 10⁵

• 1.54 x 10⁶

Multiple Choice

What is 0.000154 in scientific notation?

• 154 x 10^-3

• 15.4 x 10^-4

• 1.54 x 10^-5

• 1.54 x 10^-4

Operations with Numbers in SN

• As mentioned earlier, SN allows us to more easily perform operations (division, multiplication, addition, subtraction, exponents) on measurements.

• To perform these operations, we must first review several Laws of Exponents: the Product, Quotient, Power and Power of a Product Rule.

Product Rule

• When multiplying numbers with like bases, keep the base and add the exponents.

• For example, $10^5\times10^6=10^{5+6}=10^{11}$  

• When multiplying measurements written in SN, group the non-powers of ten in a group separate from the powers of ten.

• For example,  $5\times10^8\times6\times10^{-2}=\left(5\times6\right)\times\left(10^8\times10^{-2}\right)=30\times10^6$  

Multiple Choice

$$10^{12}\times10^{-3}$$ equals...

• $$10^{15}$$  

• $$10^9$$  

• $$10^{-4\ }$$  

• $$10^{36}$$  

Multiple Choice

$$8\times10^2\times7\times10^{-6}$$  equals...

• $$56\ \times10^{-4}$$  

• $$15\times10^{-4}$$  

• $$56\times10^{-12}$$  

• $$15\times10^{-12}$$  

Quotient Rule

• When dividing numbers with like bases, keep the base and subtract the exponents. 

• For example, $\frac{10^5}{10^6}=10^{5-6}=10^{-1}$  

• When dividing measurements written in SN, group the non-powers of ten in a group separate from the powers of ten.

• For example,  $\frac{10\times10^8}{5\times10^{-2}}=\left(\frac{10}{5}\right)\times\left(\frac{10^8}{10^{-2}}\right)=2\times10^{8-\left(-2\right)}=2\times10^{10}$  

Multiple Choice

$$\frac{10^{10}}{10^{8\ }}\$$  equals

• $$10^{-2}$$  

• $$10^{1.25}$$  

• $$10^{18}$$  

• $$10^2$$  

Multiple Choice

$$\frac{12\times10^3}{4\times10^2}$$  equals...

• $$16\times10^1$$  

• $$3\times10^5$$  

• $$3\times10^1$$  

• $$16\times10^5$$