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<!DOCTYPE html>
<html data-require="math math-format">
<head>
<meta charset="UTF-8">
<title>Multiplying and dividing scientific notation</title>
<script data-main="../local-only/main.js" src="../local-only/require.js"></script>
</head>
<body>
<div class="exercise">
<div class="vars">
<var id="DENOMINATOR">randRange(1, 9) * Math.pow(10, randRange(-5, 5))</var>
<var id="QUOTIENT">randRange(1, 99) * Math.pow(10, randRange(-5, 5))</var>
<var id="NUMERATOR">DENOMINATOR * QUOTIENT</var>
<var id="SCIENTIFIC_1">scientific(2, DENOMINATOR)</var>
<var id="SCIENTIFIC_2">scientific(2, QUOTIENT)</var>
<var id="SCIENTIFIC_3">scientific(3, NUMERATOR)</var>
<var id="MANTISSA_1">localeToFixed(scientificMantissa(2, DENOMINATOR), 1)</var>
<var id="MANTISSA_2">localeToFixed(scientificMantissa(2, QUOTIENT), 1)</var>
<var id="MANTISSA_3">localeToFixed(scientificMantissa(3, NUMERATOR), 2)</var>
<var id="MANTISSA_DIV">localeToFixed(roundTo(2, MANTISSA_3 / MANTISSA_1), 2)</var>
<var id="MANTISSA_MUL">localeToFixed(roundTo(1, MANTISSA_1 * MANTISSA_2), 1)</var>
<var id="EXPONENT_1">scientificExponent(DENOMINATOR)</var>
<var id="EXPONENT_2">scientificExponent(QUOTIENT)</var>
<var id="EXPONENT_3">scientificExponent(NUMERATOR)</var>
</div>
<div class="problems">
<div id="divide">
<p class="problem">Calculate the quotient below and give your answer in scientific notation.</p>
<p class="question">
<code>\large{\dfrac{<var>SCIENTIFIC_3</var>}{<var>SCIENTIFIC_1</var>}} =\ ?</code>
</p>
<div class="solution" data-type="multiple">
<span class="sol short30 base" data-forms="integer, decimal"><var>scientificMantissa(2, QUOTIENT)</var></span>
<code> \times 10 </code>
<sup class="sol short30 exp" data-forms="integer, decimal"><var>EXPONENT_2</var></sup>
</div>
<div class="hints">
<div>
<p>Start by collecting the significands and exponents.</p>
<p><code>
\large{\dfrac
{\blue{<var>MANTISSA_3</var>} \times \pink{10^{<var>EXPONENT_3</var>}}}
{\blue{<var>MANTISSA_1</var>} \times \pink{10^{<var>EXPONENT_1</var>}}} =
\blue{\dfrac{<var>MANTISSA_3</var>}{<var>MANTISSA_1</var>}} \times
\pink{\dfrac{10^{<var>EXPONENT_3</var>}}{10^{<var>EXPONENT_1</var>}}}}
</code></p>
</div>
<div>
<p>Then divide each term separately. When dividing exponents with the same base, subtract their powers.</p>
<p><code>= \blue{<var>MANTISSA_DIV</var>} \times \pink{10^{<var>EXPONENT_3</var> \,-\, <var>EXPONENT_1</var>}}</code></p>
<p><code>= \blue{<var>MANTISSA_DIV</var>} \times \pink{10^{<var>EXPONENT_3 - EXPONENT_1</var>}}</code></p>
</div>
<div data-if="EXPONENT_2 != EXPONENT_3 - EXPONENT_1" data-unwrap="">
<p>
To write the answer correctly in scientific notation, the first number needs to be between <code>1</code> and <code>10</code>.
In this case, we need to move the decimal one position to the right without changing the value of our answer.
</p>
<p>
We can use the fact that <code>\blue{<var>MANTISSA_DIV</var>}</code> is the same as
<code>\green{<var>MANTISSA_2</var> \div 10}</code>, or
<code>\green{<var>MANTISSA_2</var> \times 10^{-1}}</code>.
</p>
<div>
<p><code>
= \green{<var>MANTISSA_2</var> \times 10^{-1}} \times \pink{10^{<var>EXPONENT_3 - EXPONENT_1</var>}}
</code></p>
<p><code>
= <var>MANTISSA_2</var> \times 10^{\green{-1} + \pink{<var>EXPONENT_3 - EXPONENT_1</var>}}
</code></p>
<p><code>= <var>SCIENTIFIC_2</var></code></p>
</div>
</div>
</div>
</div>
<div id="multiply">
<p class="problem">Calculate the product below and give your answer in scientific notation.</p>
<p class="question"><code>
\large{\left(<var>SCIENTIFIC_2</var> \right)
\times \left(<var>SCIENTIFIC_1</var> \right) =\ ?}
</code></p>
<div class="solution" data-type="multiple">
<span class="sol short30 base" data-forms="integer, decimal"><var>scientificMantissa(3, NUMERATOR)</var></span>
<code> \times 10 </code>
<sup class="sol short30 exp" data-forms="integer, decimal"><var>EXPONENT_3</var></sup>
</div>
<div class="hints">
<div>
<p>Start by collecting the significands and exponents.</p>
<p><code>
(\blue{<var>MANTISSA_2</var>} \times \pink{10^{<var>EXPONENT_2</var>}}) \times
(\blue{<var>MANTISSA_1</var>} \times \pink{10^{<var>EXPONENT_1</var>}}) =
(\blue{<var>MANTISSA_2</var>} \times \blue{<var>MANTISSA_1</var>}) \times
(\pink{10^{<var>EXPONENT_2</var>}} \times \pink{10^{<var>EXPONENT_1</var>}})
</code></p>
</div>
<div>
<p>Then multiply each term separately. When multiplying exponents with the same base, add the powers together.</p>
<p><code>= \blue{<var>MANTISSA_MUL</var>} \times \pink{10^{<var>EXPONENT_2</var> \,+\, <var>EXPONENT_1</var>}}</code></p>
<p><code>= \blue{<var>MANTISSA_MUL</var>} \times \pink{10^{<var>EXPONENT_2 + EXPONENT_1</var>}}</code></p>
</div>
<div data-if="EXPONENT_3 != EXPONENT_1 + EXPONENT_2" data-unwrap="">
<p>
To write the answer correctly in scientific notation, the first number needs to be between <code>1</code> and <code>10</code>.
In this case, we need to move the decimal one position to the left without changing the value of our answer.
</p>
<p>
We can use the fact that <code>\blue{<var>MANTISSA_MUL</var>}</code> is the same as
<code>\green{<var>MANTISSA_3</var> \times 10}</code> or
<code>\green{<var>MANTISSA_3</var> \times 10^{1}}</code>.
</p>
<div>
<p><code>
= \green{<var>MANTISSA_3</var> \times 10^{1}} \times \pink{10^{<var>EXPONENT_2 + EXPONENT_1</var>}}
</code></p>
<p><code>
= <var>MANTISSA_3</var> \times 10^{\green{1} + \pink{<var>EXPONENT_2 + EXPONENT_1</var>}}
</code></p>
<p><code>= <var>SCIENTIFIC_3</var></code></p>
</div>
</div>
</div>
</div>
</div>
</div>
</body>
</html>
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