7.4B Square Formula and Powers with Radicals
7.4B Square Formula and Powers with Radicals
7.4B Square Formula and Powers with Radicals
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<strong>7.4B</strong> <strong>Square</strong> <strong>Formula</strong> <strong>and</strong> <strong>Powers</strong> <strong>with</strong> <strong>Radicals</strong><br />
A. <strong>Square</strong> <strong>Formula</strong><br />
Recall the square formula:<br />
<strong>Square</strong> formula:<br />
¡¢<br />
¡£<br />
<strong>Square</strong> the first term ¤¥<br />
Product times two ¦¥<br />
§¥ <strong>Square</strong> the last term<br />
After applying the square formula <strong>with</strong> radicals, remember to simplify.<br />
B. Examples<br />
Example 1: Simplify ¨©� � © §�� .<br />
Solution<br />
Apply the square formula:<br />
� � ¦ § � § ©�©<br />
� § ¤�<br />
� � ¦ ©<br />
Thus we get<br />
Ans � � �© ¦<br />
� ¦ © ¤� �<br />
� ¦ © � �¦ �<br />
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Example 2: Simplify ¨© ¦ � § � ¡�� .<br />
Solution<br />
By the square formula, we have<br />
¨ ¦ � §�� ¤¢ © ¦ � § � ¦¡<br />
Combining like terms, we have<br />
Ans ¦ � ¦� � ¤¢ © ¦ � §<br />
Example 3: Simplify ¨© � £ � © ¤ � ¦ �� .<br />
Solution<br />
By the square formula, we have<br />
Combining like terms, we have<br />
¨ � £� � ¦ © � £ © ¤ � ¦ � ¨ ¤ ¦ � �<br />
Ans ¡ � � ¦ © � £ © ¤ � ¦ OR ¡ � � ¦¤ ¨ � £� ¨ ¤ � ¦ �<br />
2
Example 4: Simplify ¨ © � ¤� © � ¤�� .<br />
Solution<br />
C. <strong>Powers</strong><br />
By the square formula, we have<br />
¨ © � ¤�� � ¦ © � ¤ © � ¤� ¨ � ¤�<br />
Bringing the power to the inside <strong>and</strong> rearranging, we have<br />
Ans ¤ ¨ � ¤�� � ¦ © � ¤ © � ¤� � ¤<br />
Recall that we use the square formula repeatedly <strong>with</strong> FOIL/factor table.<br />
Example 1: Simplify ¨© � § � © � §�¡ .<br />
Solution<br />
¨© � § © � � §�¡ ¢ ¨© � § © � � §��¨© � § © � � §�<br />
Now apply the square formula:<br />
£ ¨ � §� � ¦ © � § © � §� ¨ � §�¤ ¨© � § � © � §�<br />
Combining like terms:<br />
¨ ¦ � ¦ © � § � © §�¨© � § © � � §�<br />
Now use FOIL to get the answer:<br />
Ans ¦ © � § � ¦ © � § � ¦ ¨ � §� © � § � ¦ ¨ � §� © � §<br />
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Example 2: Simplify ¨ © � ¦� © ¦ � ¤�¡ .<br />
Solution<br />
© ¨ � ¦ � © ¦ � ¤�¡ ¢ © ¨ � ¦� © ¦ � ¤�� © ¨ � ¦� © ¦ � ¤�<br />
Now apply the square formula:<br />
£ ¨ © � ¦�� � ¦ ¨ © � ¦� ¨ © ¦ � ¤�� ¨ © ¦ � ¤��¤ ¨ © � ¦� © ¦ � ¤�<br />
Use the factor table:<br />
© ¨ � ¦�� ¦ © ¨ � ¦� © ¨ ¦ � ¤� © ¨ ¦ � ¤��<br />
© � ¦ � ¦ ¦ © ¨ � ¦�� © ¨ ¦ � ¤� © ¨ � ¦� © ¨ ¦ � ¤��<br />
� © ¦ � ¤ ¨ © � ¦�� ¨ © ¦ � ¤� ¦ ¨ © � ¦� ¨ © ¦ � ¤�� ¦ � ¤<br />
Thus we have<br />
Ans § � ¤� § ¨ © � ¦�� ¨ © ¦ � ¤�� § ¨ © � ¦� ¨ © ¦ � ¤��<br />
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